2004

1) Determine the family of orthogonal trajectories of the family $y=x+{\mathrm{ce}}^{-x}$.

[10M]

2) Show that the solution curve satisfying $(x^2-xy)y^4=y^3,$ where $y\to 1$ as $x\to 1,x\to \mathrm{\infty }$ is a conic section. Identify the curve.

[10M]

3) Solve $(1+x{)}^2{y}^{\mathrm{\prime }\mathrm{\prime }}+(1+x)y^{\mathrm{\prime }}+y=4\cos (\ln (1+x)),y(0)=1,y(e-1)=\cos 1.1$

[10M]

4) Obtain the general solution of $y^{\mathrm{\prime }\mathrm{\prime }}+2y^{\mathrm{\prime }}+2y=4e^{-x}{x}^2\sin x$

[10M]

5) Find the general solution of $({\mathrm{x}\mathrm{y}}^3+\mathrm{y})\mathrm{d}\mathrm{x}+2({\mathrm{x}}^2{\mathrm{y}}^2+\mathrm{x}+{\mathrm{y}}^4)\mathrm{d}\mathrm{y}=0$.

[10M]

6) Obtain the general solution of $({\mathrm{D}}^4+2{\mathrm{D}}^3-2\mathrm{D})\mathrm{y}=\mathrm{x}+{\mathrm{e}}^{2x}$, where $Dy={\displaystyle \frac{dy}{dx}}$.

[10M]

2003

1) Find the orthogonal trajectories of the family of eo-axial circles

where $\mathrm{g}$ is a parameter.

[10M]

2) Find three solutions of

which are linearly independent on every real interval.

[10M]

3) Solve and examine for singular solution:

[10M]

4) Solve

[10M]

5) Given $y=x$ is one solutions of

find another linearly independent solution by reducing order and write the general solation.

[10M]

6) Solve by the method of variation of parameters
${\displaystyle \frac{d^{2}y}{d{x}^2}}+a^{2}y=\sec ax,a$ is real.

[10M]

2002

1) If $(D-a{)}^4{e}^{nx}$ is denoted by $z$, prove that $z{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }n}},{\displaystyle \frac{{\mathrm{\partial }}^{2}z}{\mathrm{\partial }{n}^2}},{\displaystyle \frac{{\mathrm{\partial }}^{3}z}{\mathrm{\partial }{n}^3}}$ all vanish when $n=a$. Hence show that $e^{nx},x{e}^{nx},x^2{e}^{nx},x^3{e}^{nx}$ are all solutions of $(D-a{)}^{4}y=0$. Here $D$ stands for ${\displaystyle \frac{d}{dx}}$.

[10M]

2) Solve $4x{P}^2\cdot (3x+1{)}^2=0$ and examine for singular solutions and extraneous loci. Interpret the results geometrically.

[10M]

TBC

3)(i) Form the differential equation whose primitive is

[5M]

3)(ii) Prove that the orthogonal trajectory of system of parabolas belongs to the system itself.

[5M]

4) Using variation of parameters solve the differential equation
${\displaystyle \frac{d^{2}y}{d{x}^2}}-4x{\displaystyle \frac{dy}{dx}}+(4x^2-1)y=3e^4\sin 2x$

[10M]

5.(i) Solve the equation by finding an integrating factor of
$(x+2)\sin ydx+x\cos ydy=0$.

[5M]

5.(ii) Verify that $\varphi (x)=x^2$ is a solution of $y^{\mathrm{\prime }\mathrm{\prime }}-{\displaystyle \frac{2}{x^2}}y=0$ and find a second independent solution.

[5M]

6) Show that the solution of $(D^{2x+1}-1)y=0$, consists of ${\mathrm{A}\mathrm{e}}^x$ and $\mathrm{n}$ paris of terms of the form $e^{\alpha x}(b_r\cos \alpha x+c_r\sin \alpha x)$, where $a=\cos {\displaystyle \frac{2\pi r}{2n+1}}$ and $\alpha =\sin {\displaystyle \frac{2\pi r}{2n+1}}$, $r=1,2\dots \dots ,n$ and $b_r$, $c_r$ are arbitrary constants.

[10M]

2001

1) A tank of 100 liters capacity is initially full of water. Pure water is allowed to nun into the tank at the rate of 1 liter per minute and at the same time salt water containing ${\displaystyle \frac{1}{4}}\mathrm{k}\mathrm{g}$ of salt per liter flows into the tank at the rate of 1 liter per minute. The mixture (there is perfect mixing in the tank at all times) flows out at the rate of 2 liters per minute. Form the differential equation and find the amount of salt in the tank after $t$ minutes. Find this when $t=50$ minutes.

[10M]

2) A constant coefficient differential equation has auxiliary eqution expressible in factored form as $P(m)=m^3(m-1{)}^2{(m^2+2m+5)}^2.$ What is the order of the differential equation and find it general solution.

[10M]

3) Solve $x^2{\left({\displaystyle \frac{dy}{dx}}\right)}^2+y(2x+y){\displaystyle \frac{dy}{dx}}+y^2=0$.

[10M]

4) Using differential equations, show that the system of confocal conics given by

$\lambda$ real is self-orthogonal.

[10M]

5) Solve $(1-x^2){\displaystyle \frac{d^{2}y}{d{x}^2}}-x{\displaystyle \frac{dy}{dx}}-a^{2}y=0$ given that \(y=e^{a \sin^{-1} x\) is one solution of this equation.

[10M]

6) Find a general solution of $y^{\mathrm{\prime }\mathrm{\prime }}+y=\tan x,{\displaystyle \frac{-\pi }{2}}<x<{\displaystyle \frac{\pi }{2}}$ by variation of parameters.

[10M]

2000

1) Solve $(x^2+y^2)(1+P{)}^2-2(x+y)(1+p)(x+yp)+(x+yp{)}^2=0$ $P={\displaystyle \frac{dy}{dx}}.$ Interpret geometrically the factors in the $\mathrm{P}$. and C-discriminants of the equation $8p^{3}x=y(12p^2-9)$.

[20M]

2) Solve
(i) ${\displaystyle \frac{d^{2}y}{d{x}^2}}+{\displaystyle \frac{2}{x}}{\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{a^2}{x^4}}y=0$
(ii) ${\displaystyle \frac{d^{2}y}{d{x}^2}}+(\tan x-3\cos x){\displaystyle \frac{dy}{dx}}+2y{\cos }^{2}x={\cos }^{4}x$ by varying parameters.

[20M]