Some Substitution Methods

Category: Differential Equations, Algebra, Trigonometry

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Find the general solution for

Solution:

Consider the given equation above

Since the above equation contains trigonometric functions, then it is considered as a complicated equation with differentials. To avoid the confusion in solving the equation, it is better to rewrite the equation by using the substitution method. We will substitute all trigonometric functions with another variables.

Let

so that 

Let

so that

Substitute all the above values to the given equation, as follows

Rewrite the above equation as a first order, first degree linear equation, we have

where

and

Since the above equation is already a first order, first degree linear equation in terms of v, then the integrating factor will be equal to

Therefore, the general solution for the above equation is

but

and

and the final answer for the above equation is