Show that the solution of differential equation : $y=2(x^{2}-1)+c e^{-x^{2}}\,\mathrm{is}\,\frac{d y}{d x}+2x y-4x^{3}=0$

From the differential equation of equation y = a cos 2x + b sin 2x, where a and b are constant.

Find the sum of the order and the degree of the following differential equations : ${\frac{d^{2}y}{d x^{2}}}+{3{\sqrt\frac{{d y}}{d x}}}+(1+x)=0$

Obtain the differential equation of the family of circles passing through the points (a, 0) and (– a, 0).

Verify that ax² + by² = 1 is a solution of differential equation $x(y y_{2}+y_{1}^{2})=y y_{1}.$

Form the differential equation of all circles which is tough the x-axis at the origin.

Form the differential equation of the family of circles in the second quadrant and touching the co-ordinate axes.

Can $y=a x+{\frac{b}{a}}$ be a solution of the following differential equation ? $y=x{\frac{d y}{d x}}+{\frac{b}{\frac{d y}{d x}}}$ If no, find the solution of the D.E.

Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.

Form the differential equation representing family of curves given by (x – a)² + 2y²= a², where a is an arbitrary constant.

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x² – 3xy²) dx = (y³ – 3x²y) dy, where C is a parameter.

Obtain the differential equation of all the circles of radius r.

Find the solution of the differential equation ${\frac{d y}{d x}}=x^{3}e^{-2y}$.

Write the solution of the differential equation ${\frac{d y}{d x}}=2^{-y}.$

Find the general solution of the differential equation. $x y\,{\frac{d y}{d x}}=(x+2)(y+2).$