91Ó°ÊÓ

If \(f(x)\) is a function such that \(x \int_{0}^{x}(1-t) f(t) d t\) \(=\int_{0}^{x} t-f(t) d t\) and \(f(1)=1\) then find \(f(x)\).

Solve the following differential equations: (i) \(y^{\prime \prime}=y^{\prime}+x\) (ii) \(\mathrm{xy}^{\prime \prime}=y^{\prime} \ln \frac{\mathrm{y}^{\prime}}{\mathrm{x}}\) (iii) \(2 x y^{\prime} y^{\prime \prime}=\left(y^{\prime}\right)^{2}+1\) (iv) \(x y^{\prime \prime}+x\left(y^{\prime}\right)^{2}-y^{\prime}=0\)

Find the general solution of the linear equation of the first order \(y^{\prime}+p(x) y=q(x)\) if one particular solution, \(\mathrm{y}_{1}(\mathrm{x})\), is known.

\(\left(\mathrm{e}^{y}+1\right) \cos x d x+e^{y} \sin x d y=0\)

Show that \(y=\cos x, y=\sin x, y=c_{1} \cos x, y=c_{2} \sin x\) are all solutions of the differential equation \(\mathrm{y}_{2}+\mathrm{y}=0\)

Solve the following differential equations : (i) \(\left(2 x \cos y+y^{2} \cos x\right) d x\) \(+\left(2 y \sin x-x^{2} \sin y\right) d y=0\) (ii) \(\frac{x^{3} d x+y x^{2} d y}{\sqrt{x^{2}+y^{2}}}=y d x-x d y\)

Show that the equation \(\frac{d y}{d x}=\frac{y}{x}\) subject to the initial condition \(\mathrm{y}(0)=0\) has an infinite number of solutions of the form \(\mathrm{y}=\mathrm{Cx}\). The same equation subject to the initial condition \(\mathrm{y}(0)=\mathrm{a} \neq 0\) has no solution.

Find the curves for which \(\frac{d y}{d x}=\frac{y^{2}+3 x^{2} y}{x^{2}+3 x y^{2}}\), and determine their orthogonal trajectories.

A yeast grows at a rate proportional to its present size. If the original amount doubles in two hours, in how many hours will it triple?

Solve the following differential equations : (i) \(\left(2 x \cos y+y^{2} \cos x\right) d x\) \(+\left(2 y \sin x-x^{2} \sin y\right) d y=0\) (ii) \(\frac{x^{3} d x+y x^{2} d y}{\sqrt{x^{2}+y^{2}}}=y d x-x d y\)