The integrating factor of $x \frac{dy}{dx} + 3y = x^2$ is

Let $f$ be a non-negative function defined on $\left[0, \frac{\pi}{2}\right]$. If $\int_0^x \left(f^{\prime}(t)-\sin 2t\right) dt = \int_x^0 f(t) \tan t dt$ and $f(0)=1$, then find $\int_0^{\frac{\pi}{2}} f(x) dx$.

If the curve $y=y(x)$ is the solution of the differential equation $2(x^{2}+x^{5/4}) dy - y(x+x^{1/4}) dx = 2x^{9/4} dx, x > 0$ which passes through the point $(1, 1-\frac{4}{3} \log_{e} 2)$,then the value of $y(16)$ is equal to :

The integrating factor of the differential equation $x \cdot \frac{dy}{dx} + 2y = x^2$ is $(x \neq 0)$.

If the solution of the differential equation $x \frac{dy}{dx} + y = x e^x$ is $xy = e^x \phi(x) + C$, then $\phi(x)$ is equal to

The solution of the differential equation $(1+x) \frac{dy}{dx} - xy = 1-x$ is