Created by T. Madas
DIFFERENTIAL
EQUATIONS
(by separation of variables)
Created by T. Madas
Created by T. Madas
GENERAL
SOLUTIONS
Created by T. Madas
Created by T. Madas
Question 1 (**)
Find a general solution of the differential equation
3 y2
dy
+ 2x = 1
dx
giving the answer in the form y 3 = f ( x ) .
C4A , y 3 = A − x 2 + x
Question 2 (**)
Find a general solution of the differential equation
dy
= xy , x ≠ 0 , y ≠ 0 ,
dx
giving the answer in the form y = f ( x ) .
1 x2
y = Ae 2
Created by T. Madas
Created by T. Madas
Question 3 (**+)
Find a general solution of the differential equation
dy
= ( y + 1)(1 − 2 x ) , y ≠ −1 .
dx
giving the answer in the form y = f ( x ) .
2
y = A e x− x − 1
Question 4 (**+)
Find a general solution of the differential equation
dy
= y tan x , y > 0
dx
giving the answer in the form y = f ( x ) .
y = A sec x
Created by T. Madas
Created by T. Madas
Question 5 (**+)
Find a general solution of the differential equation
dy
= 2e x − y
dx
giving the answer in the form y = f ( x ) .
(
y = ln 2 e x + C
)
Question 6 (***)
Find a general solution of the differential equation
x2
dy
= xy + y , x ≠ 0 , y ≠ 0 ,
dx
giving the answer in the form y = f ( x ) .
[the final answer may not contain natural logarithms]
C4A , y = Ax e
Created by T. Madas
− 1x
Created by T. Madas
Question 7 (***+)
Find a general solution of the differential equation
= xy , y > 0 ,
( x2 + 3) dy
dx
giving the answer in the form y 2 = f ( x ) .
(
y 2 = A x2 + 3
)
Question 8 (***+)
Show that a general solution of the differential equation
2
dy  y 
=  ,
dx  x 
is given by
y=
x
,
1 + Ax
where A is an arbitrary constant.
proof
Created by T. Madas
Created by T. Madas
Question 9 (***+)
Find a general solution of the differential equation
dy
x ex
=
,
dx sin y cos y
giving the answer in the form f ( x, y ) = constant .
cos 2 y + 4e x ( x − 1) = C or e x ( x − 1) − sin 2 y = C or e x ( x − 1) + cos 2 y = C
Question 10 (***+)
Find a general solution of the differential equation
dy
cos 2 x = y 2 sin 2 x
dx
giving the answer in the form y = f ( x ) .
y=
Created by T. Madas
1
C + x − tan x
Created by T. Madas
Question 11 (***+)
Find a general solution of the differential equation
sec3 x
dy
= cot 2 2 y
dx
giving the answer in the form f ( x, y ) = c .
3tan 2 y − 6 y − 2sin 3 x = C
Question 12 (***+)
Find a general solution of the differential equation
e2 x
dy
= cosec2 y
dx
giving the answer in the form f ( x, y ) = c .
2 x + 2e −2 x − sin 2 y = C
Created by T. Madas
Created by T. Madas
Question 13 (****)
Show that a general solution of the differential equation
5
dy
= 2 y2 − 7 y + 3
dx
is given by
y=
A ex − 3
2 A ex −1
,
where A is an arbitrary constant.
SYN-A , proof
Created by T. Madas
Created by T. Madas
Question 14 (****+)
Show that a general solution of the differential equation
e x+2 y
dy
2
+ (1 − x ) = 0
dx
is given by
1
y = ln  2 e− x x 2 + 1 + K  ,

2 
(
)
where K is an arbitrary constant.
proof
Created by T. Madas
Created by T. Madas
Question 15
(*****)
2x
dy
= x− y+3, x > 0.
dx
Determine a general solution of the above differential equation, by using the
substitution u = y x .
SP-W , y = 1 x + 3 + Cx
3
Created by T. Madas
− 12
Created by T. Madas
Question 16
(*****)
By using the substitution y = xu , where u = f ( x ) , or otherwise, find a simplified
general solution for the following differential equation.
x
dy
= 2 x 2 + 2 xy + y .
dx
SP-J , y = Ax e− x − x
Created by T. Madas
Created by T. Madas
Question 17
(*****)
Use differentiation to find a simplified general solution for the following differential
equation.
(
2
 dy 
 dy 
x − 1   − 2 xy   + y 2 = 1 .
 dx 
 dx 
2
)
SP-F ,
(A+ y
Created by T. Madas
)
x 2 − 1 ( y + Bx + C ) = 0
Created by T. Madas
Question 18 (*****)
A circle touches the x axis at the origin O .
It is further given that the equation of such a circle satisfies the differential equation
= y f ( x) ,
( x2 − y 2 ) dy
dx
for some function f .
Use an algebraic method to find an expression for f ( x ) .
SP-W , f ( x ) = 2 x
Created by T. Madas
Created by T. Madas
Question 19 (*****)
The non zero functions u ( x ) and v ( x ) satisfy the integral equations

u ( x ) dx = ux 2
and




u ( x ) v ( x ) dx =  u ( x ) dx   v ( x ) dx  .





Determine, in terms of an arbitrary constant, a simplified expression for u ( x ) and a
2
similar expression for v ( x )  .
−1
Ae x
2
B
SP-L , u ( x ) = 2 , v ( x )  =
2
x
(1 − x ) 1 − x 2
(
Created by T. Madas
)
Created by T. Madas
Question 20
(*****)
The positive solution of the quadratic equation x 2 − x − 1 = 0 is denoted by φ , and is
commonly known as the golden section or golden number.
g φ
a) Show, with a detailed method, that F ( x ) = f (φ ) x ( ) is a solution of the
differential equation,
F ′ ( x ) = F −1 ( x ) ,
where f and g are constant expressions of φ , to be found in simplified form.
b) Verify the answer obtained in part (a) satisfies the differential equation, by
differentiation and function inversion.
[You may assume that F ( x ) is differentiable and invertible]
1
 1 φ
SPX-L , F ( x ) =   xφ = φ1−φ xφ
φ 
Created by T. Madas
Created by T. Madas
SPECIFIC
SOLUTIONS
Created by T. Madas
Created by T. Madas
Question 1 (**)
Solve the differential equation
dy 4 x
+
= 0, y ≠ 0,
dx y
subject to the condition y = 2 at x = 0 .
Give the answer in the form f ( x, y ) = constant.
4 x2 + y 2 = 4
Question 2 (**)
Solve the differential equation
dy cos 2 x
=
,
dx
y
subject to the condition y = 6 at x =
π
4
x > 0, y > 0 ,
, giving the answer in the form y 2 = f ( x ) .
y 2 = 35 + sin 2 x
Created by T. Madas
Created by T. Madas
Question 3 (**)
Solve the differential equation
dy
= 6 xy 2 ,
dx
with y = 1 at x = 2 , giving the answer in the form y = f ( x ) .
y=
1
13 − 3x 2
Question 4 (**)
Solve the differential equation
dy 3sin 3 x
=
,
dx
y
subject to the condition y = 3 at x =
π
3
, giving the answer in the form y 2 = f ( x ) .
y 2 = 7 − 2 cos 3 x
Created by T. Madas
Created by T. Madas
Question 5 (**)
Solve the differential equation
dy 2 x
=
,
dx
y
with y = 2 at x = 1 , giving the answer in the form y 2 = f ( x ) .
y 2 = 2 x2 + 2
Question 6 (**)
Solve the differential equation
dy
10
,
=
dx ( x + 1)( x + 2 )
subject to the condition y = 0 at x = 0 , giving the answer in the form y = f ( x ) .
y = 10ln
Created by T. Madas
2x + 2
x+2
Created by T. Madas
Question 7 (**)
Solve the differential equation
( )
1
dy cos 3 x
=
,
dx
y
subject to the condition y = 1 at x =
π
2
, giving the answer in the form y 2 = f ( x ) .
( )
y 2 = 6sin 1 x − 2
3
Question 8
(**)
Solve the differential equation
dy
= 3x 2 y
dx
subject to the condition y = 0 at x = 1 , giving the answer in the form y = f ( x ) .
(
)
y = 1 x3 − 1
4
Created by T. Madas
2
Created by T. Madas
Question 9 (**)
Solve the differential equation
dy
=
dx
y
,
x +1
y ≠ 0 , x ≠ −1 ,
subject to the condition y = 9 at x = 8 , giving the answer in the form y = f ( x ) .
y = x +1
Question 10 (**)
Solve the differential equation
dy
= 2 x 2 y − 1,
dx
y>1
2
subject to the condition y = 1 at x = 0 , giving the answer in the form y = f ( x ) .
2
(
)
y = 1 x4 + 1
2
Created by T. Madas
Created by T. Madas
Question 11 (**)
Solve the differential equation
dy
+ y 2 e x = 0,
dx
subject to the condition y = 1 at x = 0 , giving the answer in the form y = f ( x ) .
2
y=
1
x
e +1
Question 12 (**)
dy 2 y
=
, x > 0 , y > 0.
dx
x
Show that the solution of the above differential equation subject to the boundary
condition y = 3 at x = 1 , is given by
y = 3x 2 .
proof
Created by T. Madas
Created by T. Madas
Question 13 (**+)
Solve the differential equation
dy
= yx 2 , x ≠ 0 , y ≠ 0 ,
dx
subject to the condition y = 1 at x = 1 , giving the answer in the form y = f ( x ) .
1
y = e3
Created by T. Madas
( x3 −1)
Created by T. Madas
Question 14 (**+)
Solve the differential equation
dy
= y2 x , x ≠ 0 , y ≠ 0 ,
dx
with y = −2 at x = 1 .
Give the answer in the form y =
A
3
, where A and B are integers.
1 + Bx 2
SYN-N , y =
6
3
1− 4x 2
Created by T. Madas
Created by T. Madas
Question 15
(**+)
Solve the differential equation
x3
dy
= 2 y2
dx
subject to the condition y = 1 at x = 1 , giving the answer in the form y = f ( x ) .
2
y=
x2
x2 + 1
Question 16 (**+)
Solve the differential equation
dy x − y
+e
=0,
dx
subject to y = 0 at x = 0 , giving the answer in the form f ( x, y ) = constant .
ex + e y = 2
Created by T. Madas
Created by T. Madas
Question 17 (**+)
Solve the differential equation
dy
= xy e x , x > 0 , y > 0 ,
dx
subject to boundary condition y = e at x = 1 .
Give the answer in the form ln y = f ( x ) .
ln y = x e x − e x + 1
Question 18
(**+)
Solve the differential equation
4 y +1
dy
=−
dx
x2
subject to the condition, y = 2 at x = 2 , giving the answer in the form y = f ( x ) .
3
y=
Created by T. Madas
1
x
2
−
1
4
Created by T. Madas
Question 19 (**+)
Solve the differential equation
dy 2 y 2
= 3
dx
x
subject to the condition y = −1 at x = 1 , giving the answer in the form y = f ( x ) .
y=
Created by T. Madas
x2
1 − 2 x2
Created by T. Madas
Question 20 (**+)
Given that y = 2 at x = 0 , solve the differential equation
dy
= 4 + y2 ,
dx
giving the answer in the form y = f ( x ) .
You may assume that

1
2
a +x
2
dx =
1
x
arctan   + C .
a
a
π

y = 2 tan  2 x + 
4

Question 21 (***)
( x + 1)
dy
= 3y , y > 0 .
dx
Solve the differential equation subject to the condition y = 16 at x = 1 , to show that
3
y = 2 ( x + 1) .
proof
Created by T. Madas
Created by T. Madas
Question 22 (***)
Solve the differential equation
dy
y
=
, y>0
dx x ( 2 − x )
subject to the condition y = 1 at x = 1 , giving the answer in the form y 2 = f ( x ) .
C4M , y 2 =
x
2− x
Question 23 (***)
Find the solution of the differential equation
dy
= y sin x , y > 0
dx
subject to the condition y = 10 at x = π , giving the answer in the form y = f ( x ) .
y = 10e−1−cos x
Created by T. Madas
Created by T. Madas
Question 24 (***)
Solve the differential equation
x2
dy
= y 2 − 3x 4 y 2
dx
subject to the condition y = 1 at x = 1 , giving the answer in the form y = f ( x ) .
2
C4K , y =
x
4
x +1
Question 25 (***)
Solve the differential equation
dy
= 4 yx3 ,
dx
y≠0
subject to the condition y = 1 at x = 1 , giving the answer in the form y = f ( x ) .
4
y = e x −1
Created by T. Madas
Created by T. Madas
Question 26 (***)
Solve the differential equation
= y ( x + 1) ,
(1 − x2 ) dy
dx
y ≠ 0, x ≠ ±1 ,
subject to the condition y = 2 at x = 1 , giving the answer in the form y = f ( x ) .
2
y=
1
1− x
Question 27 (***)
Solve the differential equation
dy 2 x ln x
=
, x >0, y >0
dx
y
subject to the condition y = 2 e at x = e , giving the answer in the form y 2 = f ( x ) .
y 2 = x 2 ( 2ln x − 1) + 3e2
Created by T. Madas
Created by T. Madas
Question 28 (***)
Solve the differential equation
dy
= 4 xy − 3 yx 2
dx
subject to the condition y = 1 at x = 2 , giving the answer in the form y = f ( x ) .
2
3
SYN-I , y = e 2 x − x
Question 29
(***)
Solve the differential equation
dy 10 − y
=
dx
5
subject to the condition y = 1 at x = 0 , giving the answer in the form y = f ( x ) .
y = 10 − 9e
Created by T. Madas
− 15 x
Created by T. Madas
Question 30 (***)
Show that the solution of the differential equation
2 y −1
dy
=
,
dx
x2
x ≠ 0, y > 1
2
subject to the condition y = 1 at x = 1 , is given by
y=
5x2 − 4 x + 1
2 x2
.
proof
Created by T. Madas
Created by T. Madas
Question 31 (***+)
Solve the differential equation
x ( x + 2)
dy
= y, x > 0, y > 0
dx
subject to the condition y = 2 at x = 2 , giving the answer in the form y 2 = f ( x ) .
MP2-N , y 2 =
Created by T. Madas
8x
x+2
Created by T. Madas
Question 32 (***+)
Solve the differential equation
dy
5y
=
dx ( 2 + x )(1 − 2 x )
subject to the condition y = 2 at x = 0 , giving the answer in the form y = f ( x ) .
C4E , y =
2+ x
1 − 2x
Question 33 (***+)
Solve the differential equation
dy
y
=
, y > 0 , x > −1
dx ( x + 1)( x + 3)
subject to the condition y = 2 at x = 1 , giving the answer in the form y 2 = f ( x ) .
y2 =
Created by T. Madas
8 ( x + 1)
x+3
Created by T. Madas
Question 34 (***+)
( 3x + 2 )( x + 3)
dy
= 7 y , y > 0 , x > −3 .
dx
Show that the solution of the above differential equation subject to the boundary
condition, y = 6 at x = 4 , is given by
y=
3 ( 3x + 2 )
.
x+3
proof
Question 35 (***+)
Solve the differential equation
ey
dy
+ x ex = 0 , x < 1
dx
subject to the condition y = 0 at x = 0 , giving the answer in the form y = f ( x ) .
y = x + ln (1 − x )
Created by T. Madas
Created by T. Madas
Question 36 (***+)
Solve the differential equation
( 2 x − 3)( x − 1)
dy
= y ( 2 x − 1) , y ≠ 0 ,
dx
subject to the condition y = 10 at x = 2 , giving the answer in the form y = f ( x ) .
10 ( 2 x − 3)
C4I , y =
x −1
2
Question 37 (***+)
Solve the differential equation
ex
subject to y =
π
4
dy
x
=
, 0< y <π ,
dx sin 2 y
at x = 0 , giving the answer in the form cos f ( y ) = g ( x ) .
cos 2 y = 2 x e− x + 2e − x − 2
Created by T. Madas
Created by T. Madas
Question 38 (***+)
Solve the differential equation
y (13 − 2 x )
dy
=
, y ≠0,
dx ( 2 x − 3)( x + 1)
subject to the condition y = 4 at x = 2 , giving the answer in the form y = f ( x ) .
y=
108 ( 2 x − 3)
2
( x + 1)3
Question 39 (***+)
Solve the differential equation
dy
= yx 2 cos x, x > 0, y > 0
dx
subject to the condition y = 1 at x = π , giving the answer in the form ln y = f ( x ) .
ln y = x 2 sin x + 2 x cos x − 2sin x + 2π
Created by T. Madas
Created by T. Madas
Question 40 (***+)
2y
dy
1
=
, y ≠0,
dx x + 3
Show that the solution of the above differential equation, subject to the boundary
condition y = 1 at x = 1 , can be written as
y 2 = ln
e ( x + 3)
.
4
proof
Question 41
(***+)
dy
cos 2 4 x = y , y > 0 .
dx
Show that the solution of the above differential equation subject to the boundary
condition y = e3 at x =
π
16
is given by
1 (11+ tan 4 x )
y = e4
.
C4G , proof
Created by T. Madas
Created by T. Madas
Question 42 (***+)
Show that if y = a at t = 0 , the solution of the differential equation
dy
= ω a2 − y2
dt
(
)
1
2
,
where a and ω are positive constants, can be written as
y = a cos ωt .
You may assume that

 x
dx = arcsin   + C .
a
a2 − x2
1
proof
Question 43 (***+)
Solve the differential equation
yey
2
dy
= e2 x , x ≠ 0 , y ≠ 0 ,
dx
subject to the condition y = 2 at x = 2 , giving the answer in the form y 2 = f ( x ) .
y2 = 2x
Created by T. Madas
Created by T. Madas
Question 44 (***+)
1
( y − 2 )( y + 1)
≡
P
Q
+
, y ≠ −1, 2
( y − 2 ) ( y + 1)
a) Find the value of each of the constants P and Q .
b) Hence, show that the solution of the differential equation
dy
= x 2 ( y − 2 )( y + 1)
dx
can be written as
3
y−2
= A e x , where A is a constant.
y +1
c) Given further that y = 5 when x = 0 , show clearly that
y=
4 + ex
3
2 − ex
3
.
P = 1, Q = −1
3
3
Created by T. Madas
Created by T. Madas
Question 45 (***+)
Solve the differential equation
dy
y
=
,
dx ( 2 x + 1)( x + 1)
1
x>− , y>0
2
subject to the condition y = 2 at x = 0 , giving the answer in the form y = f ( x ) .
y=
4x + 2
x +1
Question 46 (***+)
Solve the differential equation
dy
y
=
,
dx x ( x + 1)2
subject to the condition y =
x > 0, y > 0
1
at x = 1 , giving the answer in the form ln y = f ( x ) .
2
1
1
 x 
ln y = ln 
−
+
 x +1 x +1 2
Created by T. Madas
Created by T. Madas
Question 47 (***+)
Solve the differential equation
dy
= 3 x e3 x + y
dx
subject to the condition y = 0 at x = 0 , giving the answer in the form e y = f ( x ) .
ey =
3
3x
e − 3 x e3 x + 2
Question 48 (***+)
−5
dy
= 2 y − 150 , y < 75
dx
Solve the above differential equation, given that when x = 0 , y = 275 .
Give the answer in the form y = f ( x ) .
2
− x
C4B , y = 75 + 200e 5
Created by T. Madas
Created by T. Madas
Question 49 (***+)
(1 + x2 ) dydx = x (1 + y ) , with y = 0 at x = 0 .
Show that the solution of the above differential equation is
(
y = 1 + x2
)
1
2
−1.
proof
Created by T. Madas
Created by T. Madas
Question 50
(***+)
x
dy
= y ( y + 1) , x > 0 , y > 0
dx
Show that the solution of the above differential equation subject to the boundary
condition y = 1 at x = 1 , is given by
2
3
y=
x
.
1− x
MP2-L , proof
Created by T. Madas
Created by T. Madas
Question 51 (***+)
x
dy
= y 1 + 2 x2 , x > 0 , y > 0
dx
(
)
Show that the solution of the above differential equation subject to y = e at x = 1 , is
2
y = x ex .
proof
Created by T. Madas
Created by T. Madas
Question 52 (****)
dy y 2 − 1
=
, x >0, y >0
dx
x
Show that the solution of the above differential equation subject to y = 2 at x = 1 , is
y=
3 + x2
3 − x2
.
SYN-E , proof
Created by T. Madas
Created by T. Madas
Question 53 (****)
ex
dy
+ y 2 = xy 2 , x > 0 , y > 0
dx
Show that the solution of the above differential equation subject to y = e at x = 1 , is
y=
1 x
e .
x
MP2-B , proof
Created by T. Madas
Created by T. Madas
Question 54 (****)
Solve the differential equation
dy
= x 2 e3 y − x
dx
f y
subject to the condition y = 0 at x = 0 , giving the answer in the form e ( ) = g ( x ) .
(
)
e −3 y = 3e − x x 2 + 2 x + 2 − 5
Question 55 (****)
Solve the differential equation
5
1 dy
= 2 x 2 + 1 cos 2 2 y
x dx
(
subject to x = 0 , y =
π
8
)
, giving the answer in the form tan f ( y ) = g ( x )
6


tan 2 y = 1  2 x 2 + 1 + 11
12 

(
Created by T. Madas
)
Created by T. Madas
Question 56 (****)
Solve the differential equation
( 2 x + 1) −
dy
( 2 x − 1)3 = 0
dx
subject to the condition y = 0 at x = 0 , giving the answer as y = f ( x ) .
y=−
Created by T. Madas
x
( 2 x − 1)2
Created by T. Madas
Question 57 (****)
dy
xy
= 2
, x, y > 2
dx x − 3 x + 2
Solve the differential equation above, subject to the boundary condition y = 1 at
3
x = 3 , to show that
2
2 ( x − 2)
y=
.
3 ( x − 1)
proof
Created by T. Madas
Created by T. Madas
Question 58
(****)
Solve the differential equation
dy
= x sin 2 x cos 2 y
dx
subject to the condition x =
π
4
, y = 0 , giving the answer in the form tan y = f ( x ) .
tan y = 1 ( sin 2 x − 2 x cos 2 x − 1)
4
Question 59 (****)
( x − 1)
dy
= 2x y
dx
Solve the differential equation above, subject to the boundary condition y = 4 at
x = 2 , to show that
e
y
= e x ( x − 1) .
proof
Created by T. Madas
Created by T. Madas
Question 60 (****)
dy
sec x = y 2 − y
dx
Solve the differential equation above, subject to the boundary condition, y = 1 at
2
x = 0 , to show that
y=
1
1 + esin x
.
MP2-E , proof
Created by T. Madas
Created by T. Madas
Question 61 (****)
Solve the differential equation
y (1 − x )
dy
=
,
dx (1 + x ) 1 + x 2
(
)
subject to the condition x = 0 , y = 1 , giving the answer in the form y 2 = f ( x ) .
y
2
2
1+ x)
(
=
1 + x2
Question 62 (****)
Solve the differential equation
(1 + x2 ) dydx = x (1 − y 2 ) ,
y ≠ ±1 ,
subject to the condition y = 0 at x = 0 , giving the answer in the form y = f ( x ) .
y=
Created by T. Madas
x2
x2 + 2
Created by T. Madas
Question 63 (****)
(1 + x )
dy
= y (1 − x ) , y > 0 , x > −1 .
dx
Solve the above given differential equation, subject to the boundary condition y = 1 at
x = 0 , to show that
y = ( x + 1) e − x .
2
MP2-Z , proof
Created by T. Madas
Created by T. Madas
Question 64 (****+)
Solve the differential equation
dy
= 24cos 2 y cos3 x
dx
subject to the condition y =
π
4
at x =
π
6
, giving the answer in the form tan y = f ( x ) .
tan y = 24sin x − 8sin 3 x − 10
Created by T. Madas
Created by T. Madas
Question 65 (****+)
Solve the differential equation
2
dy
= 2− 2 ,
dx
y
subject to the condition y = 2 at x = 1 , giving the answer in the form x = f ( y ) .
3y − 3
x = 1 y + 1 ln
2
4
y +1
Created by T. Madas
Created by T. Madas
Question 66 (****+)
xy + (1 + x )
dy
= y.
dx
Solve the differential equation subject to the condition y = 3 at x = 0 , to show that
y = 3 (1 + x ) e − x .
2
proof
Created by T. Madas
Created by T. Madas
Question 67 (****+)
dy
cot x = 1 − y 2 .
dx
Solve the differential equation above, subject to the boundary condition y = 0 at
x=
π
4
, to show that
y=
1 − 2cos 2 x
1 + 2cos 2 x
.
SYN-H , proof
Created by T. Madas
Created by T. Madas
Question 68 (****+)
( x + 2)
dy
+ y ( x + 1) = 0 , x > −2 .
dx
Solve the differential equation above, subject to the boundary condition y = 2 at
x = 0 , to show that
y = ( x + 2 ) e− x .
proof
Created by T. Madas
Created by T. Madas
Question 69
(****+)
A curve y = f ( x ) satisfies the differential equation
y = 1−
dy
x +1
, y > 1 , x > −1
dx ( x − 1)( x + 2 )
a) Solve the differential equation to show that
ln ( y − 5 ) +
1 2
x + 4 x − 2ln ( x + 1) = C .
2
When x = 0 , y = 2 .
b) Show further that
2
2 − 12 x
.
y = 1 + ( x + 1) e
proof
Created by T. Madas
Created by T. Madas
Question 70
(****+)
Solve the differential equation
50
dy
= 20 − y ,
dx
given that when x = 0 , y = 0 , giving the answer in the form x = f ( y ) .
x = 2000 ln
Created by T. Madas
20
− 100 y
20 − y
Created by T. Madas
Question 71
(****+)
Solve the differential equation
d2y
dx
2
+2
dy
= 1,
dx
dy
given that y = − 1 and
= 1 at x = 0 , giving the answer in the form y = f ( x ) .
4
dx
MP2-W , y = 1  2 x − e−2 x 

2
Created by T. Madas
Created by T. Madas
Question 72
(****+)
A curve y = f ( x ) satisfies the differential equation
dy k ( 9 − x )
=
, y > 0, 0 ≤ x ≤ 9 ,
dx
y
where k is a positive constant.
dy
It is further given that y = 1 ,
= 2 at x = 1 .
2 dx
Find the possible values of x when
dy 1
= .
dx 5
SYN-V , x = 5 ∪ x = 13
Created by T. Madas
Created by T. Madas
Question 73
(****+)
A curve passes through the point with coordinates 1 , log 2 ( log 2 e )  and its gradient
function satisfies
dy
= 2y , x ∈ , x < 2 .
dx
Find the equation of the curve in the form y = f ( x )
X , y = − log 2 ( 2 − x ) ln 2 
Created by T. Madas
Created by T. Madas
Question 74
(*****)
The function y = f ( x ) satisfies the differential equation
2 xy ( y + 1)
dy
=
,
dx sin 2 x + 1 π
6
(
)
subject to the condition y = 1 at x = 0 .
Find the exact value of y when x =
π
12
.
1
MP2-S , y =
e
Created by T. Madas
1π
6
−1
Created by T. Madas
Question 75
(*****)
Determine, in the form y = f ( x ) , a simplified solution for the following differential
equation.
dy
cos x + 4 y 2 sin x = sin x ,
dx
y = 15 at x = 1 π .
34
3
SP-H , y =
Created by T. Madas
sec4 x − 1
(
)
2 sec 4 x + 1
Created by T. Madas
Question 76
(*****)
The function v = f ( t ) satisfies the differential equation
dv
1 1 
= k − ,
dt
v h
where k and h are non zero constants.
Given that v = h − 1 at t = 0 , solve the differential equation to show that
k t +v
(h − v)eh
=A
where A is a non zero constant.
SYN-X , proof
Created by T. Madas
Created by T. Madas
Question 77
(*****)
The function y = f ( x ) satisfies the differential equation
dy
1 1 
= k − ,
dx
h x
where k and h are non zero constants.
It is further given that y = −1 ,
dy
d2y
= −1 ,
= 2 at x = −1 .
dx
dx 2
Solve the differential equation to show that
2
e y−x = ( y + 2) .
SPX-F , proof
Created by T. Madas
Created by T. Madas
Question 78
(*****)
The function y = f ( x ) satisfies the differential equation
d
dy d 2
yx 2 =
x , x>0
dx
dx dx
( )
( )
subject to the condition y = 4 at x = 3 .
Find a simplified expression for y = f ( x ) .
SP-I , y =
Created by T. Madas
4
( x − 2 )2
Created by T. Madas
Question 79
(*****)
Use appropriate techniques to solve the following differential equation.
d2y
dx
2
=−
144
y
3
,
y ( 0) = 6 ,
dy
dx
= 0.
x =0
x2 y2
SPX-J ,
+
=1
9 36
Created by T. Madas
Created by T. Madas
Question 80
(*****)
Solve the differential equation
d2y
2
 dy 
+ 4  = 1,
2
dx
 dx 
given that y = 0 and
dy 1
=
at x = 0 , giving the answer in the form y = f ( x ) .
dx 6
1 1 + 2 e 4 x  1
SP-T , y = ln 
− x
4  3  2
Created by T. Madas
Created by T. Madas
Question 81 (*****)
Solve the following differential equation
d2y
2
 dy 
= x  ,
2
dx
 dx 
given further y = 0 ,
dy
= 2 at x = 0 .
dx
Give the answer in the form x = f ( y ) .
SP-N , x =
Created by T. Madas
e y −1
( )
= tanh 1 x
2
e +1
y
Created by T. Madas
Question 82 (*****)
dy
=
dx
y4 − y2
x4 − x2
, x > 0 , y > 0.
Find the solution of the above differential equation subject to the boundary condition
2
y=
at x = 2 .
3
Give the answer in the form y =
2x
, where f ( x ) is a function to be found.
f ( x)
MP2-T , f ( x ) = 3 + x 2 − 1
Created by T. Madas
Created by T. Madas
Question 83
(*****)
2
d 2 y  dy 
+   =1.
dx 2  dx 
Given that y =
dy
= 0 at x = 0 , show that
dx
y = − x + ln  1 1 + e2 x  .
2

(
)
SP-B , proof
Created by T. Madas
Created by T. Madas
Question 84 (*****)
The non zero function f ( x ) satisfies the integral equation
 f ( x ) dx = 
f ( x ) dx ,
f (0) = 1 .
4
2
 dy 
Use the substitution f ( x ) =   , to find a simplified expression for f ( x ) .
 dx 
SP-K , f ( x ) = 1 e 4 x
4
Created by T. Madas
Created by T. Madas
Question 85
(*****)
It is required to sketch the curve with equation y = f ( x ) , defined over the set of real
numbers, in the greatest domain.
The curve has the property that at every point on the curve, the second derivative
equals to the first derivative squared.
Showing all the relevant details, sketch the graph of y = f ( x ) , given further that the
curve passes through the point ( 0, 2 ) and the gradient at that point is 1.
SPX-O , graph
Created by T. Madas
Created by T. Madas
Question 86
(*****)
It is given that a function with equation y = f ( x ) is a solution of the following
differential equation.
2
(1 − x2 ) ddx2y − 2 x dydx + y = 0 .
Show with a clear method that

x
1
(
)
f ( x ) dx = x 2 − 1 f ′ ( x ) .
SP-X , proof
Created by T. Madas
Created by T. Madas
Question 87
(*****)
Solve the following differential equation
2
y
d2y
dy
dy
 dy 
+   + 2 y = 0 , y ( 0) = 2 ,
( 0 ) = − 12 .
2
dx
dx
dx
 dx 
Give the answer in the form y 2 = f ( x ) .
SPX-G , y 2 = 3 + e−2 x
Created by T. Madas
Created by T. Madas
Question 88
(*****)
The function with equation y = f ( x ) satisfies the differential equation
d2y
dx
2
=
2  dy 
1 −  ,
2 x − 1  dx 
y ( 0) = 1,
dy
( 0 ) = −1 .
dx
Solve the above differential equation giving the answer in the form y = f ( x ) .
SPX-B , y = x + 1 + ln 2 x − 1
Created by T. Madas
Created by T. Madas
Question 89
(*****)
The positive solution of the quadratic equation x 2 − x − 1 = 0 is denoted by φ , and is
commonly known as the golden section or golden number.
g φ
a) Show, with a detailed method, that F ( x ) = f (φ ) x ( ) is a solution of the
differential equation,
F ′ ( x ) = F −1 ( x ) ,
where f and g are constant expressions of φ , to be found in simplified form.
b) Verify the answer obtained in part (a) satisfies the differential equation, by
differentiation and function inversion.
[You may assume that F ( x ) is differentiable and invertible]
1
 1 φ
V , SPX-L , F ( x ) =   xφ = φ1−φ xφ
φ 
Created by T. Madas