Differential Equations

The solution of the differential equation $(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^2$ is

If $\frac{d y}{d x}-y \log _e 2=2^{\sin x}(\cos x-1) \log _e 2$, then $y$

The integrating factor of $\frac{x d y}{d x}-y=x^4-3 x$ is

The number of solutions of $\frac{d y}{d x}=\frac{y+1}{x-1}$, when $y(1)=2$ is

$\left(\frac{d y}{d x}\right) \tan x=y \sec ^2 x+\sin x$, find general solution

The general solution of the differential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^{2}\right) d x$ is

If the slope of the tangent to the curve at any point $\mathrm{P}(x, y)$ is $\frac{y}{x}-\cos ^{2} \frac{y}{x}$, then the equation of a curve passing through $\left(1, \frac{\pi}{4}\right)$ is

The solutions of $(x+y+1) d y=d x$ are

The solution of the differential equation $(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^{2}$ is

If ${f}^{'}(x)=f(x)+{∫}_{0}^{1}f(x)dx, f(0)=1$, then $f(x)=$

Solution of the equation $\frac{dy}{dx}={e}^{x-y}({e}^{x}-{e}^{y})$ is

The solution of the differential equation $\frac{{d}^{2}y}{d{x}^{2}}=\sin3x+{e}^{x}+{x}^{2}$ when ${y}_{1}(0)=1$ and $y(0)=0$, is