plz use differential equation method to solve this:

latex: y'=y+\frac{x}{y}; \: \: y\left(0\right)=1

Step-by-step explanation:

We have the differential equation with initial conditions .

First, notice that the equation can be rewritten as

,

which is a Bernoulli equation. Once we have recognized the type of the equation we know how to continue. Recall that a Bernoulli equation has the general form

.

In this particular case we have . This kind of equation is solved by the change of variable . In our exercise we get . Now we take derivatives and get

which es equivalent to .

Then, we substitute the value of we have obtained in the original equation:

.

The next step is to multiply the whole equation by , in order to eliminate the denominator of . Thus,

.

Recall that , then

.

This last equation is a linear equation, which has general solution

.

So, let us calculate the integral that appear in the formula:

.

Then, the solution for is

.

Now, we return the change of variable:

.

The last step is to find the value of the constant . In order to do this, substitute the initial value:

.

Thus, we have the equation

that gives .

Therefore,

.