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.

Find the general solution of the following differential equation : $y-x{\frac{d y}{d x}}=a\left(y^{2}+{\frac{d y}{d x}}\right).$

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

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

Find the particular solution of the following differential equation. $\cos y\,d x+(1+2e^{-x})\sin y\,d y=0;y(0)={\frac{\pi}{4}}$

Solve the following differential equation : $\mathrm{cosec}\,x\,\mathrm{log}\,y\,\frac{d y}{d x}+x^{2}y^{2}=0\,.$

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.

Find the particular solution of the differential equation ${\frac{d y}{d x}}\ =1+x+y+x y$ given that y = 0 when x = 1.

Find the particular solution of the differential equation $\log\left(\frac{d y}{d x}\right)=3x+4y,$ given that y = 0, when x = 0.

Solve the differential equation $(x^{2}\ -\ y x^{2})d y$ + $(y^{2}+x^{2}y^{2})d x=0,$ given that y = 1 when x = 1.

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}\;=1+x^{2}+y^{2}+x^{2}y^{2},$ given that y = 1, when x = 0.