Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

Show that the equation $x=a\cos (nt+b)+\frac{k}{n^2-p^2}\sin pt$, satisfies the differential equation $\frac{d^{2}x}{d{t}^2}+n^{2}x=k\sin pt$.

Obtain the differential equation whose general solution is $y=a\sin \left(bx+c\right)$, where $a,b \text{and} c$ are arbitrary constants.

Find the differential equation of all those parabolas whose foci are at the origin and axes are along the $x-$axis.

Form the differential equation of the family of parabolas whose axes are parallel to $x-$axis.

Obtain the differential equation of family of ellipses whose two foci lie on the $x-$axis and whose centre is at the origin.

Form the differential equation of family of hyperbolas whose centre is at the origin and foci lie on the $x-$axis.

Show that $y=a{e}^{2x}+b{e}^x+2\sin x$ is the general solution of the differential equation $y_2-3y_1+2y+2\left(3\cos x-\sin x\right)=0$. Hence solve it, under the initial conditions, $y_2-3y_1+2y+2\left(3\cos x-\sin x\right)=0; y=4, y_1=7$ when $x=0$.

Show that the general solution of the differential equation $y_2+4y=0$ is $y=a\sin 2x+b\cos 2x$. Hence solve the boundary value problem $y_2+4y=0;$ $y=0$ when $x=\frac{\mathrm{\pi }}{8}$ and $y=1$ when $x=\frac{\mathrm{\pi }}{6}$.