shadsiei

2021-02-25

Solve the linear equations by considering y as a function of x, that is, y = y(x).

$\frac{dy}{dx}-y=4{e}^{x},y\left(0\right)=4$

2abehn

Skilled2021-02-26Added 88 answers

Variation of parameters

First, solve the linear homogeneous equation by separating variables. Rearranging terms in the equation gives

Now, the variables are separated, x appears only on the right side, and y only on the left.

Integrate the left side in relation to y, and the right side in relation to x

which is

By taking exponents, we obtain

Hence,we obtain

where

Next, we need to find the particular solution

Therefore, we consider

Let's assume that

since

Therefore

which gives

Now, we can find the function u :

Since we need to find only one function that will male this work, we don’t need to introduce the constant of integration c. Hence,

Recall that

The general solution is

Integrating Factor technique

This equation is linear with

Hence,

So, an integrating factor is

and the general solution is