2008

1) Show that the functions $y_1(x)=x^2$ and $y_2(x)=x^2{\log }_{e}x$ are linearly independent. Obtain the differential equation that has $y_1(x)$ and $y_2(x)$ as the independent solutions.

[10M]

2) Solve the following ordinary differential equatjon of the second degree:
$y{\left({\displaystyle \frac{dy}{dx}}\right)}^2+(2x-3){\displaystyle \frac{dy}{dx}}-y=0$

[10M]

3) Reduce the equation $(x{\displaystyle \frac{dy}{dx}}-y)(x-y{\displaystyle \frac{dy}{dx}})=2{\displaystyle \frac{dy}{dx}}$ to Clairaut’s form and obtain thereby the singular integral of the above equation.

[10M]

4) Solve:
$(1+x{)}^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+(1+x){\displaystyle \frac{dy}{dx}}+y=4\cos {\log }_e(1+x)$

[10M]

5) Find the general solution of the equation
${\displaystyle \frac{d^{2}y}{d{x}^2}}-\cot x{\displaystyle \frac{dy}{dx}}-(1-\cot x)y=e^x\sin x$

[10M]

6) Use the method of variation of parameters to solve the differential equation
$x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}-y=x^2{e}^x$.

[10M]

2007

1) Find the orthogonal trajectories of the family of the curves ${\displaystyle \frac{x^2}{a^7}}+{\displaystyle \frac{y^2}{b^2+\lambda }}=1$, $\lambda$ being a parameter.

[10M]

2) Show that $e^{2x}$ and $e^{3x}$ are linearly independent solutions of

Find the general solution when $y(0)=0$ and ${\displaystyle \frac{dy}{dx}}=1$ at $x=0$.

[10M]

3) Find the family of curves whose tangents form an ${\displaystyle \frac{\pi }{4}}$ angle with the byperbola $xy=C$.

[10M]

4) Apply. the method of variation of parameters to solve

[10M]

5) Solve

By using the method of removal of first derivate.

[10M]

6) Find the general solution of
$(1-x^2){\displaystyle \frac{d^{2}y}{d{x}^2}}-2x{\displaystyle \frac{dy}{dx}}+3y=0,$ if $y=x$ is a solution of it

[10M]

2006

1) From $x^2+y^2+2ax+2by+c=0,$ derive differential equation not containing $a,b$ or $c$.

[10M]

2) Discuss the solution of the differential equation $y^2[1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2]=a^2$.

[10M]

3) Solve
$x{\displaystyle \frac{d^{2}y}{d{x}^2}}+(1-x){\displaystyle \frac{dy}{dx}}-y=e^x$

[10M]

4) Solve
${\displaystyle \frac{d^{4}y}{d{x}^4}}-y=x\sin x$

[10M]

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

[10M]

6) Reduce
$xy{\left({\displaystyle \frac{dy}{dx}}\right)}^2-(x^2+y^2+1){\displaystyle \frac{dy}{dx}}+xy=0$ to Clairaut’s form and find its singular solution.

[10M]

2005

1) Form the differential equation that represents all parabolas each of which has latus rectum $4\mathrm{a}$ and whose axes are parallel to the $x-axis$.

[10M]

2.(i) The auxiliary polynomial of a certain homogeneous linear differential equation with constant coefficients in factored form is $P(m)=m^4(m-2{)}^6{(m^2-6m+25)}^3$ What is the order of the differential equation and write a general solution?

[5M]

2.(ii) Find the equation of the one parameter family of parabolas given by $y^2=2cx+c^2$, $c$ real and show that this family is self-orthogonal.

[5M]

3) Solve and examine for singular solution the following equation.

$p^2(x^2-a^2)-2pxy+y^2-b^2=0$.

[10M]

4) Solve the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}+9y=\sec 3x$.

[10M]

5) Given $y=x+{\displaystyle \frac{1}{x}}$ is one solution, solve the differential equation $x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+x{\displaystyle \frac{dy}{dx}}-y=0$ by reduction of order method.

[10M]

6) Find the general solution of the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}-2y{\displaystyle \frac{dy}{dx}}-3y=2e^x-10\sin x$ by the method of undetermined coefficients.

[10M]