Question

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

Solve $\left(1+x^{2}\right){\frac{d y}{d x}}+2x y-4x^{2}=0$ subject to the initial condition y(0) = 0.

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}-y=\cos x\ {\mathrm{~for~}}x=0,y=1.$

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

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 equations of the tangent and normal to the curve x = a sin³θ, y = b cos³θ at $\mathbf{\theta}={\frac{\pi}{4}}\,.$