1988

1) Solve the differential equation

2) Show that the equation $(12x+7y+1)dx+(7x+4y+1)dy=0$ represents a family of curves having as asymptotes the lines $3x+2y-1=0,2x+y+1=0$.

3) Obtain the differential equation of all circles in a plane in the form ${\displaystyle \frac{d^{3}y}{d{x}^3}}\{1+{\left({\displaystyle \frac{dy}{dx}}\right)}^2\}$-$3{\displaystyle \frac{dy}{dx}}{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^2=0$

1987

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

2) If $f(t)=t^{p-1},g(t)=t^{q-1}$ for $t>0$ but $f(t)=g(t)=0$ for $t^{\mathrm{\prime }\mathrm{\prime }}0,$ and $h(t)=f^{\ast }g,$ the convolution of $f$, $g$, show that $h(t)={\displaystyle \frac{\mathrm{\Gamma }(p)\mathrm{\Gamma }(q)}{\mathrm{\Gamma }(p+q)}}{t}^{p+q-1};t\ge 0$ and $\mathrm{p},\mathrm{q}$ are positive constants. Hence, deduce the formula

1985

1) Consider the equation $y^{\mathrm{\prime }}+5y=2$. Find that solution $\varphi$ of the cquation which satisfics $\varphi (1)=3{\varphi }^{\mathrm{\prime }}(0)$.

2) Use Laplace transform to solve the differential equation $x^{\mathrm{\prime }\mathrm{\prime }}-2x^{\mathrm{\prime }}+x=e^{\mathrm{\prime }},(‘={\displaystyle \frac{d}{dt}})$ such that

3) For two functions $\mathrm{f},\mathrm{g}$ both absolutely integrable on $(-\mathrm{\infty },\mathrm{\infty }),$ define the convolution $f^{\ast }g$. If $\mathrm{L}(\mathrm{f}),\mathrm{L}(\mathrm{g})$ are the Laplace transforms of $\mathrm{f},\mathrm{g}$ show that $L\left(f^{\ast }g\right)=L(f)\cdot L(g)$.

4) Find the Laplace transform of the function

1984

1) Solve ${\displaystyle \frac{d^{2}y}{d{x}^2}}+y=\sec x$.

2) Using the transformation $y={\displaystyle \frac{u}{x^k}},$ solve the equation $x{\mathrm{y}}^{\mathrm{\prime }}+(1+2\mathrm{k}){\mathrm{y}}^{\mathrm{\prime }}+\mathrm{x}\mathrm{y}=0$.

3) Solve the equation $(D^2+1)x=t\cos 2t$ given that $x_0=x_1=0$ by the methodof Laplace transform.

1983

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

2) Solve $(y^2+yz)dx+(xz+z^2)dy+(y^2-xy)dz=0$.

3) Solve the equation by the method of Laplace transform, given that $y=-3$ when $t=0,y=-1$ when $t=1$.