`y' = 5x/y` Solve the differential equation

The general solution of a differential equation in a form of `y'=f(x,y)` can be evaluated using direct integration. The derivative of y denoted as `y'` can be written as `(dy)/(dx)` then `y'= f(x)` can be expressed as `(dy)/(dx)= f(x)` .

For the problem: `y'=5x/y` , we let ` y'=(dy)/(dx) ` to set it up as:

`(dy)/(dx)= 5x/y`

Cross-multiply `dx` to the right side:

`(dy)= 5x/ydx`

Cross-multiply y to the left side:

`ydy=5xdx`

Apply direct integration on both sides:

`int ydy=int 5xdx`

Apply basic integration property:` int c*f(x)dx = c int f(x) dx` on the right side.

`int ydy=int 5xdx`

`int ydy=5int xdx`

Apply Power Rule for integration: `int u^n du= u^(n+1)/(n+1)+C` on both sides.

For the left side, we get:

`int y dy = y^(1+1)/(1+1)`

            `= y^2/2`

For the right side, we get:

`int x dx = x^(1+1)/(1+1)+C`

            `= x^2/2+C`

Note: Just include the constant of integration "C" on one side as the arbitrary constant of a differential equation.

Combining the results from both sides, we get the general solution of the differential equation as:

`y^2/2=x^2/2+C`

or` y =+-sqrt(x^2/2+C)`