Differentiation summary

Chapter 9: Differential Equations
Solving Differential Equations:
1. Direct Integration
Differential
Equation
Solution

dy
 f  x dx y   f  x  dx  C

dy
 f  y dx 1 dy f  y  dx



1
 f  y  dy 
1

 f  y  dy

d2 y
 f  x dx 2

1

 1 dx

dy dx 


F x  C



y

 f  x  dx  C
 F  x   C dx

 G  x   Cx  D

 xC

2. Substitution
Use the substitution v  x  y to find the general solution of the differential equation

dy
2
  x  y . dx Step 1: Apply product rule/quotient rule/chain rule to

v



differentiate the given substitution with respect to x and express

dv dx dy dx  1

dv dy in terms of x, v,
.
dx dx Step 2: Replace

dy dv by in the differential equation. dx dx

Step 3: Replace all terms in y by terms in v and x . Simplify and express

dv in terms of either x or v (but not both). dx 

dy

dx dv 1  dx x y dy dx

dv
1
dx

 x  y

2

 x  y

2

dv
2
1   x  y  dx dv
1  v2 dx dv
 v2  1 dx Step 4: Apply direct integration.

Step 6: Replace all terms in v by terms in y and x .

dv dx 1 dv
2
v  1 dx
1
 v 2  1 dv tan 1  v 

 1


 1 dx

 xC

tan 1  v 

 xC

1

 xC

tan
Step 7: Express y in terms of x .

 v2  1

 x  y

tan 1  x  y   x  C y  tan  x  C   x

Note:


A particular solution of the given differential equations can be found if initial condition(s) is provided.



If the initial condition(s) is not given, and different values are given to the arbitrary constant, then a family of solution curves will be obtained from the general solution of the given differential equation.