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Lecture

MAT136H1 Lecture Notes - Product Rule, Integrating Factor


Department
Mathematics
Course Code
MAT136H1
Professor
all

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9.5 Differential Equations
Linear Equations
Question #4 (Medium): Solving the Initial Value for the Linear Differential Equation
Strategy
Solving initial value means once the linear differential equation has been solved, given the x and y value
combination, the arbitrary constant factor can be determined by working out the equation backward.
Sample Question
Solve the initial value problem.
  ,  , 
Solution
First the equation need to be arranged in the form of linear differential equation:
 
  
Then
. The integrating factor is:     
  

 
Multiply the integrating factor to the linear differential equation on both sides:

  
;

  
; the left side is then in the form of product rule: 
  
; then integrate
both sides:
  
 
 
 
The equation can be divided by the integrating factor:  
 
Now since , plug into the equation    and    to find the missing value :

  

;  
 ; then  
Therefore the solution to the initial value problem is  
 
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