1994

1) Solve: ${\displaystyle \frac{dy}{dx}}+x\sin 2y=x^3{\cos }^2$

[10M]

2) Show that if ${\displaystyle \frac{1}{Q}}[{\displaystyle \frac{\mathrm{\partial }p}{\mathrm{\partial }y}}-{\displaystyle \frac{\mathrm{\partial }Q}{\mathrm{\partial }x}}]$ is a function of $x$ only, say, $f(x),$ then $F(x)=e^{\int f(x)dx}$ is an integrating factor of $Pdx+Qdy=0$

[10M]

3) Find the family of curves whose tangent from an angle ${\displaystyle \frac{\pi }{4}}$ with the hyperbola $xy=c$.

[10M]

4) Transform the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}\cos x+{\displaystyle \frac{dy}{dx}}\sin x-2y{\cos }^{3}x=2{\cos }^{5}x$ into one having $z$ as independent variable where $z=\sin x$ and solve it.

[10M]

5) If ${\displaystyle \frac{d^{2}x}{d{t}^2}}+{\displaystyle \frac{g}{b}}(x-a)=0(a,b$ and $g$ being positive constants) and $x=a$, and ${\displaystyle \frac{dx}{dt}}=0$ when $t=0$, show that $x=a+(a^{\mathrm{\prime }}-a)\cos \sqrt{{\displaystyle \frac{g}{b}}t}$

[10M]

6) Solve $(D^2-4D+4)y=8x^2{e}^{2x}\sin 2x$ where $D\equiv {\displaystyle \frac{d}{dx}}$

[10M]

1993

1) Determine the curvature for which the radius of curvature is proportional to the slope of the tangent.

[10M]

2) Show that the system of confocal conics ${\displaystyle \frac{x^2}{a^2+\lambda }}+{\displaystyle \frac{y^2}{b^2+\lambda }}=1$ is self-orthogonal.

[10M]

3) Solve $\{y(1+1/x)+\cos y\}dx+\{x+\log x-x\sin y\}dy=0$

[10M]

4) Solve $y{\displaystyle \frac{d^{2}y}{d{x}^2}}-2{\left({\displaystyle \frac{dy}{dx}}\right)}^2=y^2$.

[10M]

5) Solve ${\displaystyle \frac{d^{2}y}{d{t}^2}}+{\omega }_0^{2}y=a\cos \omega t$ and discuss the nature of solution as $\omega \stackrel{\text{hello}}{\to }{\omega }_0$.

[10M]

6) Solve $(D^4+D^2+1)y{e}^{-x/2}\cos (x\sqrt{3}/2)$.

[10M]

1992

1) By eliminating the constants $a$, $b$ obtain the differential equation of which $\mathrm{x}\mathrm{y}={\mathrm{a}\mathrm{e}}^{\mathrm{x}}+{\mathrm{b}\mathrm{e}}^{-\mathrm{x}}+{\mathrm{x}}^2$ is a solution.

[10M]

2) Find the orthogonal trajectories of the family of the semi-cubical parabolas ay ${}^2={\mathrm{x}}^3$, where $a$ is a variable parameter.

[10M]

3) Show that

$(4x+3y+1)dx+(3x+2y+1)dy=0$ represents hyperbolas having the following lines as asymptotes

[10M]

4) Solve the following differential equation:

[10M]

5) Find the curves for which the portion of $y-axis$ cut off between the origin and the tangent varies as the cube of the abscissa of the point of contact.

[10M]

6) Solve the following differential equation:

$({\mathrm{D}}^2+4)y=\sin 2\mathrm{x}$, given that when $\mathrm{x}=0,$ then $\mathrm{y}=0$ and ${\displaystyle \frac{dy}{dx}}=2$

[10M]

7) Solve: $({\mathrm{D}}^3-1)\mathrm{y}={\mathrm{x}\mathrm{e}}^x+{\cos }^2\mathrm{x}$

[10M]

8) Solve: $(x^2{D}^2+xD-4)y=x^2$.

[10M]

1991

1) If the equation $\mathrm{M}\mathrm{d}\mathrm{x}+\mathrm{N}\mathrm{d}\mathrm{y}=0$ is of the form ${\mathrm{f}}_1$ (xy). $\mathrm{y}\mathrm{d}\mathrm{x}+{\mathrm{f}}_2(\mathrm{x}\mathrm{y}).\mathrm{x}\mathrm{d}\mathrm{y}=0,$ then show that ${\displaystyle \frac{1}{Mx-Ny}}$.

2) Solve the differential cquation. $(x^2-2x+2y^2)dx+2xydy=0$.

3) Given that the differential equation $(2x^2{y}^2+y)dx$ $-(x^{3}y-3x)dy=0$ has an integrating factor of the form $x^h{y}^{2=k},$ find its general solution.

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

5) Solve the differential equation

6) Solve the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}-4{\displaystyle \frac{dy}{dx}}-5y=x{e}^{-x},$ given that $y=0$ and ${\displaystyle \frac{dy}{dx}}=0,$ when $\mathrm{x}=0$.

1990

1.(a) If the equation ${\lambda }^n+a_1{\lambda }^{n-1}+\dots \dots \dots +a_n=0$ (in unknown $\lambda$ ) has distinct roots ${\lambda }_1,{\lambda }_2,\dots \dots \dots \dots {\lambda }_n$. Show that the constant coefficients of differential equation
${\displaystyle \frac{d^{n}y}{d{x}^n}}+a_1{\displaystyle \frac{d^{n-1}y}{d{x}^{n-1}}}+\dots \dots \dots +a_{n-1}{\displaystyle \frac{dy}{dx}}+a_n=b$

has the most general solution of the form $y=c_0(x)+c_1{e}^{2,x}+c_2{e}^{2x}+\dots \dots \dots \dots ..+c_2{e}^{\lambda x}$, where $c_1,c_2\dots \dots .c_n$ are parameters. What is $c_0(x)$?

1.(b) Analyses the situation where the $\lambda$-equation in (a) has repeated roots.

2) Solve the differential equation $x^2{\displaystyle \frac{d^{2}y}{d{x}^2}}+2x{\displaystyle \frac{dy}{dx}}+y=0$ is explicit form. If your answer contains imaginary quantities, recast it in a form free of those.

3) Show that if the function ${\displaystyle \frac{1}{t-f(t)}}$ can be integrated (w.r.t ‘t’), then one can solve ${\displaystyle \frac{dy}{dx}}=f(y/x),$ for any given $f$. Hence or otherwise, show that

4) Verify that $y={({\sin }^{-1}x)}^2$ is a solution of $(1-x^2){\displaystyle \frac{d^{2}y}{d{x}^2}}-x{\displaystyle \frac{dy}{dx}}=2.$ Find also the most general solution.

1989

1) Find the value of $y$ which satisfies the equation $(x{y}^3-y^3-x^2{e}^x)+3x{y}^2{\displaystyle \frac{dy}{dx}}=0$; given that $y=1$ when $x=1$.

2) Prove that the differential equation of all parabolas lying in a plane is ${\displaystyle \frac{d}{dx}}{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^{-2/3}=0$.

3) Solve the differential equation

${\displaystyle \frac{d^{3}y}{d{x}^3}}-{\displaystyle \frac{d^{2}y}{d{x}^2}}-6{\displaystyle \frac{dy}{dx}}=1+x^2$.