Find the differential equation of the family of curves y²= 4ax.

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.

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

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{1+y^{2}}{1+x^{2}}}$ given that y(0) = ${\sqrt{3}}.$

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

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

Find the integrating factor of the differential equation ${\frac{d y}{d x}}+y\ =\ {\frac{1+y}{x}}$