Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to:

Let us consider a curve $y=f(x)$ passing through the point $(-2, 2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x)+x f'(x)=x^2$. Then:

The integrating factor of $\frac{dy}{dx} + y = \frac{1+y}{x}$ is

If the solution curve $y=f(x)$ of the differential equation $(x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0$ for $x>2$ passes through the point $(3, 15)$, then the local maximum value of $f$ is:

Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x) = \int_{0}^{x} f(t) \, dt$. Then, the value of $f(\log_{e} 5)$ is:

If $y=\sin x+A \cos x$ is the general solution of $\frac{dy}{dx}+f(x)y=\sec x$, then an integrating factor of the differential equation is