Question

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

Answer 1

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

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

$\frac{dy}{dx}+Py=Q$

$P=\frac{2x}{1+x^2},Q=\frac{\cos x}{1+x^2}$

$IF=e^{\int Pdx}$

$=e^{\int \frac{2x}{1+x^2}dx}$

$\therefore IF=1+x^2$

$y\times IF=\int Q\times IFdx+c$

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

$y\times (1+x^2)=\int \cos xdx+c$

$y(1+x^2)=\sin x+c$