To reduce the differential equation $\frac{dy}{dx} + P(x)y = Q(x)y^n$ to the linear form,the substitution is

The general solution of the differential equation $(x+2y^3) \frac{dy}{dx} - y = 0, y > 0$ is

Let $f: R \rightarrow R$ be a continuous function satisfying $f(x)=x+\int_0^x f(t) dt$,for all $x \in R$. Then,the number of elements in the set $S=\{x \in R: f(x)=0\}$ is

Let $f$ be a differentiable function with $\lim _{x \rightarrow \infty} f(x)=0$. If $y^{\prime}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$, then:

Integrating factor of $x \frac{dy}{dx} - y = x^4 - 3x$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)((y+1)e^{x^{2}/2}-x)$,with $y(2)=0$. Then $y'(1)$ is equal to . . . .