1988

1) Solve the differential equation

\[\dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}=2 e^{x} \sin x\]

---

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

---

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

1987

1) Solve the equation \(x \dfrac{d^{2} y}{d x^{2}}+(1-x) \dfrac{d y}{d x}=y+e^{x}\).

---

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

\[B(p, q)=\dfrac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}\]

1985

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

---

2) Use Laplace transform to solve the differential equation \(x^{\prime \prime}-2 x^{\prime}+x=e^{\prime},\left(‘=\dfrac{d}{d t}\right)\) such that

\[x(0)=2, x^{\prime}(0)=-1\]

---

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

---

4) Find the Laplace transform of the function

\[f(t)=\left\{\begin{array}{rl}1 & 2 n \pi \leq t<(2 n+1) \pi \\ -1 & (2 n+1) \pi \leq t \leq(2 n+2) \pi\end{array}\right.\] \[\mathbf{n}=0,1,2, \ldots \ldots \ldots \ldots\]

1984

1) Solve \(\dfrac{d^{2} y}{d x^{2}}+y=\sec x\).

---

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

---

3) Solve the equation \(\left(D^{2}+1\right) x=t \cos 2 t\) given that \(x_{0}=x_{1}=0\) by the methodof Laplace transform.

1983

1) Solve \(x \dfrac{d^{2} y}{d x}+(x-1) \dfrac{d y}{d x}-y=x^{2}\).

---

2) Solve \(\left(y^{2}+y z\right) d x+\left(x z+z^{2}\right) d y+\left(y^{2}-x y\right) d z=0\).

---

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