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 d3ydx3{1+(dydx)2}-3dydx(d2ydx2)2=0

1987

1) Solve the equation xd2ydx2+(1−x)dydx=y+ex.

2) If f(t)=tp−1,g(t)=tq−1 for t>0 but f(t)=g(t)=0 for t′′0, and h(t)=f∗g, the convolution of f, g, show that h(t)=Γ(p)Γ(q)Γ(p+q)tp+q−1;t≥0 and p,q are positive constants. Hence, deduce the formula

1985

1) Consider the equation y′+5y=2. Find that solution ϕ of the cquation which satisfics ϕ(1)=3ϕ′(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

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

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.