Math 3120
Differential Equations
with
Boundary Value Problems
Chapter 2:
First-Order Differential Equations
Section 2-5: Solutions By Substitution
Homogenous Equations
If a function f possesses the property for some real number ,
f (tx, ty)  t  f ( x, y)
then f is said to be a homogenous functions of degree α.
For example,
y 2  yx
y3  x3
f ( x, y ) 
x
2
,
A first-order DE in differential form
M ( x, y )dx  N ( x, y )dy  0
f ( x, y ) 
x2 y
(1)
is said to be homogenous if both coefficient functions M and N
are homogenous equations of the same degree.
Note: By using substitution ( y = ux or x =vy) we can reduce a
homogenous equation to a separable first-order DE.
Method to solve Homogenous Equation
Check if the DE is an homogenous equation.
Write out the substitution y = ux
Solve the new equation (which is always separable) to find u.
Through the substitution
u
y
x
go back to the old function y.
Example 1: Pg 74 Q2
Solve the given differential equation by using an appropriate
substitution.
( x  y )dx  xdy  0
Example 2:
Pg 74 Q4
Solve the given differential equation by using an appropriate
substitution.
ydx  2( x  y )dy
Example 3:
Pg 74 Q11
Solve the given initial-value problem.
dy
xy
 y3  x3 ,
dx
2
y (1)  2
Bernoulli’s Equation
The differential equation
dy
 P( x) y  f ( x) y n
dx
(2)
where n is any real number is called Bernoulli’s equation.
Note
For n=0 and n=1, equation (2) is linear.
For n ≠ 0 and n ≠ 1, the substitution u = y1-n reduces eq (2) to a linear
equation.
Method to solve Bernoulli’s Equation
Check if the DE is a Bernoulli’s equation. Find the parameter n
from the differential equation.
Write out the substitution u = y1-n
Solve the new linear equation to find v
Through the substitution y = u(1/1-n) go back to the old function y.
Example 4:
Solve the given differential equation by using an appropriate
substitution.
dy
 y  ex y2
dx
Example 5:
Solve the given initial-value problem.
dy
x
 2 xy  3 y 4 ,
dx
2
1
y (1) 
2
Reduction To Separation Variables
The differential equation
dy
 f ( Ax  By  C )
dx
(3)
can always be reduced to an equation with separable variables
by means of the substitution u = Ax + By + C, B ≠ 0.
Example 6:
Solve the given differential equation by using an appropriate
substitution.
dy
 ( x  y  1) 2
dx