MATH 212 WORKSHEET 1
1.3 Classification of Differential Equations
1. State the order of the given ordinary differential equations. Determine whether
the equation is linear or nonlinear.
′′
d2 y
ii) y 2 =
dx
′
i) (2 − t)y + 2ty = cos t,
iii) y 3 y (4) + y = cos x,
s
1+(
dy 2
),
dx
iv) x2 R(6) + 2eR = 0.
2. Verify that the given function y = ϕ(x) or y = ϕ(t) is an explicit solution of
the given first order differential equation.
i) (y − x)y ′ = y − x + 8,
ii) 2
dy
= y 3 cos t,
dt
′
iii) y + 2xy = 1,
√
ϕ(x) = x + 4 x + 2,
1
ϕ(t) = (1 − sin t)− 2
ϕ(x) = e
−x2
Z x
0
z2
2
e dz + C1 e−x .
3. Find an ordinary differential equation satisfied by the function f (y) such
that ϕ(x, y) = f (y) sin x is a solution of the partial differential equation
uxx + uyy = 0.
2.1 Linear Equations; Method of Integrating Factors
4. Find the general solutions of the following linear first order differential
equations:
dy
C ln t
+
, ii) t − y = t2 sin t, Ans. y = Ct − t cos t,
t
t
dt
2
t
t 11
iii) y ′ = 2y + t2 + 5, Ans. y = Ce2t − − − ,
2
2
4
t+1
iv) (t2 − 1)y ′ + 2y = (t + 1)2 , Ans. y = (C + t)(
).
t−1
i) t2 y ′ + ty = 1, Ans. y =
5. Solve the given initial value problems.
dy
1
5
6t − 2
= 2t − 3y, y(0) = , Ans. y = e−3t +
,
dt
3
9
9
dy
ex + 2 − e
ii) x + y = ex , y(1) = 2, Ans. y =
.
dx
x
i)
6. If the general solution of y ′′ + 4y = 0 is given by y(t) = C1 sin(2t) + C2 cos(2t),
then solve the initial value problem:
y ′′ + 4y = 0
π
π
y( ) = 1, y ′ ( ) = 1,
4
4
Ans. y(t) = sin(2t) −
1
cos(2t).
2
2.2 Separable Equations
7. Solve the following initial value problems.
dy
1
= y − y 2 , y(0) = − ,
dx
3
dy
= y − y 2 , y(−1) = 2,
ii)
dx
i)
ex
,
ex − 4
2ex+1
Ans.y = x+1
.
2e
−1
Ans.y =
8. Solve the given differential equation by separation of variables.
q
√
i) x 1 + y 2 dx = y 1 + x2 dy,
q
√
Ans. 1 + x2 = 1 + y 2 + C,
dy
6 cos2 (3x)
cos3 (3x) = 0, Ans.y =
,
dx
C cos2 (3x) + 1
dy
xy + 3x − y − 3
iii)
=
, Ans.y − 5 ln(y + 3) = x − 5 ln(x + 4) + C,
dx
xy − 2x + 4y − 8
dy
1
iv) (ex + e−x )
= y 2 , Ans.y = −
.
dx
arctan(ex ) + C
ii) y 2 sin(3x) +