Question

(2ax+x^2)dy/dx = a^2+2ax solve this differential equations

Answer 1

Variable separable
\[\frac{dy}{dx}=\frac{a^2+2 a x}{x (2 a+x)}\]
\[y\text{ = }\int\limits \frac{a^2+2 a x}{x (2 a+x)} \, dx\]

Answer 2

Using partial fractions you will get
\[\int\limits \left(\frac{a}{2 x}+\frac{3 a}{2 (2 a+x)}\right) \, dx\]
Then integrate as usual.

Answer 3

in second step i doesn't understand

Answer 4

You mean you don't know how to get
\[y\text{ = }\int\limits\limits \frac{a^2+2 a x}{x (2 a+x)} \, dx ~~~~~?\]

Answer 5

ya

Answer 6

From here
\[(2ax+x^2)\frac{dy}{dx} = a^2+2ax\]
Divide both sides by \(2ax+x^2\)
\[\frac{dy}{dx} = \frac{a^2+2ax}{(2ax+x^2)}\]
Multiply dx
\[dy = \frac{a^2+2ax}{(2ax+x^2)}dx\]
\[y = \int \frac{a^2+2ax}{(2ax+x^2)}dx\]

Answer 7

can you tell me about partial fractions