If $y'' - 3y' + 2y = 0$ where $y(0) = 1$ and $y'(0) = 0$, then the value of $y$ at $x = \log_{e} 2$ is

The solution of $(2y - x) \frac{dy}{dx} = 1$ is

The integrating factor of the first order differential equation $x^{2}(x^{2}-1) \frac{dy}{dx} + x(x^{2}+1)y = x^{2}-1$ is

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$,suppose $f^{\prime}(x)=3 f(x)+\alpha$,where $\alpha \in \mathbb{R}, f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{e} 3\right)$ is equal to ............

Find the general solution of the differential equation: $x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x$.

If the function $y = f(x)$ satisfies the differential equation $(x^3 + 1)dy = x(1 - 3xy)dx$ and $f(0) = 0$,then $\mathop {\lim }\limits_{x \to 0} \frac{x^2}{f(x)}$ is equal to