First Order Linear Equation, 2

Category: Differential Equations, Trigonometry, Integral Calculus

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Find the particular solution for

when

Solution:

Consider the given equation 

If 

then

If

then

Since

then the given equation is not an Exact Equation. We need to find the integrating factor first and then multiply it on both sides of an equation so that the given equation becomes an Exact Equation. 

Consider again the given equation

Arrange the above equation into its standard form as follows

where

and

The integrating factor is

The general solution for the above equation is 

If

then

Therefore, the particular solution for the above equation is 

We can consider the above equation as a final answer but if you want to further simplify the above equation, consider the Half Angle Formula as follows

Hence, the above equation becomes