2000

1) Show that $3{\displaystyle \frac{d^{2}y}{d{x}^2}}+4x{\displaystyle \frac{dy}{dx}}-8y=0$ has an integral which is a polynomial in $x$. Deduce the general solution.

[10M]

2) Reduce ${\displaystyle \frac{d^{2}y}{d{x}^2}}+P{\displaystyle \frac{dy}{dx}}+Qy=R,$ where, $P,Q,R$ are functions of $x$, to the normal form. Hence solve ${\displaystyle \frac{d^{2}y}{d{x}^2}}-4x{\displaystyle \frac{dy}{dx}}+(4x^2-1)y=-3e^{x^2}\sin 2x$

[10M]

3) Solve the differential equation $y=x-2ap+a{p}^2.$ Find the singular solution and interpret it geometrically.

[10M]

4) Show that $(4x+3y+1)dx+(3x+2y+1)dy=0$ represents a family of hyperbolas with a common axis and tangent at the vertex.

[10M]

5) Solve $x{\displaystyle \frac{dy}{dx}}-y=(x-1)({\displaystyle \frac{d^{2}y}{d{x}^2}}-x+1)$ by the method of variation of parameters.

[10M]

1999

1) Solve the differential equation ${\displaystyle \frac{xdx+ydy}{xdy-ydx}}=({\displaystyle \frac{1-x^2-y^2}{x^2+y^2}}{)}^{1/2}$

[10M]

2) Solve ${\displaystyle \frac{d^{3}y}{d{x}^3}}-3{\displaystyle \frac{d^{2}y}{d{x}^2}}+4{\displaystyle \frac{dy}{dx}}-2y=e^x+cosx$

[10M]

3) By the method of variation of parameters, solve the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}+a^{2}y=sec(ax)$.

[10M]

1998

1) Solve the differential equation: $xy-{\displaystyle \frac{dy}{dx}}=y^3{e}^{-x^2}$

[10M]

2) Show that the equation: $(4x+3y+1)dx+(3x+2y+1)dy=0$ represents a family of hyperbolas having as asymptotes the lines $x+y=0,2x+y+1=0$.

[10M]

3) Solve the differential equation: $y=3px+4p^2$

[10M]

4) Solve the differential equation: ${\displaystyle \frac{d^{2}y}{d{x}^2}}-5{\displaystyle \frac{dy}{dx}}+6y=e^{4x}(x^2+9)$

[10M]

5) Solve the differential equation: ${\displaystyle \frac{d^{2}y}{d{x}^2}}+2{\displaystyle \frac{dy}{dx}}+y=xsinx$

[10M]

6) Solve the differential equation: $x^3{\displaystyle \frac{d^{3}y}{d{x}^3}}+2x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+2y=10(x+{\displaystyle \frac{1}{x}})$.

[10M]

1997

1) Solve the initial value problem

[10M]

2) Solve $(x^2-y^2+3x-y)dx+(x^2-y^2+x-3y)dy=0$.

[10M]

3) Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is $3\mathrm{m}\mathrm{m}$, and one hour later has been reduced to $2\mathrm{m}\mathrm{m}$. find an expression for the radius of the rain drop at any time.

[10M]

4) Solve ${\displaystyle \frac{d^{4}y}{d{x}^4}}+6{\displaystyle \frac{d^{3}y}{d{x}^3}}+11{\displaystyle \frac{d^{2}y}{d{x}^2}}6{\displaystyle \frac{dy}{dx}}=20e^{-2x}\sin x$

[10M]

5) Make use of the transformation $y(x)=u(x)\sec x$ to obtain the solution of ${\mathrm{y}}^{\mathrm{\prime }\mathrm{\prime }}-2{\mathrm{y}}^{\mathrm{\prime }}\tan \mathrm{x}+5\mathrm{y}=0,{\mathrm{y}}^{\mathrm{\prime }}(0)=0,{\mathrm{y}}^{\mathrm{\prime }}(0)=\sqrt{6}$
[10M]

6) Solve $\begin{array}{c}(1+2x{)}^2{\displaystyle \frac{d^{2}y}{d{x}^2}}-6(1+2x){\displaystyle \frac{dy}{dx}}+16y=8(1+2x{)}^2\\ y(0)=0,y^{\mathrm{\prime }}(0)=2\end{array}$

[10M]

1996

1) Find the curves for which the sum of the reciprocals of the radius vector and polar subtangent is constant.

[8M]

8.(b) Solve: $x^2(y-px)=y{p}^2,p={\displaystyle \frac{dy}{dx}}$

[15M]

2) $y\sin 2xdx-(1+y^2+{\cos }^{2}x)dy=0$

[15M]

3) ${\displaystyle \frac{d^{2}y}{d{x}^2}}+2{\displaystyle \frac{dy}{dx}}+10y+37\sin 3x=0$ Find the value of $y$ when $x=\pi /2,$ if it is given that $y=3$ and ${\displaystyle \frac{dy}{dx}}=0,$ when $x=0$

[15M]

4) Solve: ${\displaystyle \frac{d^{4}y}{d{x}^4}}+2{\displaystyle \frac{d^{3}y}{d{x}^3}}-3{\displaystyle \frac{d^{2}y}{d{x}^2}}=x^2+3e^{2x}+4\sin x$

[15M]

5) Solve: $x^3{\displaystyle \frac{d^{3}y}{d{x}^3}}+3x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}+y=x+\log x$

[15M]

1995

1) Determine a family of curves for which the ratio of the y-intercept of the tangent to the radius vector is a constant.

[20M]

2) Solve $(2x^2+3y^2-7)xdx-(3x^2+2y^2-8)ydy=0$

[20M]

3) Test whether the equation $(x+y{)}^{2}dx-(y^2-2xy-x^2)dy=0$ is exact and hence solve it.

[20M]

4) Solve $x^3{\displaystyle \frac{d^{3}y}{d{x}^3}}+2x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+2y=10(x+{\displaystyle \frac{1}{x}})$

[20M]

5) Determine all real valued solutions of the equation $y^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\cdot {\mathrm{i}\mathrm{y}}^{\mathrm{\prime }\mathrm{\prime }}+{\mathrm{y}}^{\mathrm{\prime }}\cdot \mathrm{i}\mathrm{y}=0,y^{\mathrm{\prime }}={\displaystyle \frac{dy}{dx}}$

[20M]

6) Find the solution of the equation $y^{\mathrm{\prime }\mathrm{\prime }}+4y=8\cos 2x$ give that $y=0$ and $y^{\mathrm{\prime }}=2$ when $x=0$

[20M]