Question

Find the general solution of each of the following differential equations:
$(1+x^2)\frac{dy}{dx}=xy$.

Answer 1

$(1+x^2)\frac{dy}{dx}=xy$

$\int \frac{1}{y}dy=\frac{1}{2}\int \frac{2x}{(1+x^2)}dx$

$\Rightarrow \log |y|-\frac{1}{2}\log |1+x^2|=\log |C|$

$\Rightarrow \log \left|\frac{y}{\sqrt{1+x^2}}\right|=\log |C|\Rightarrow \frac{y}{\sqrt{1+x^2}}=\pm C=C_1$.