1994

1) Solve: \(\dfrac{d y}{d x}+x \sin 2 y=x^{3} \cos ^{2}\)

[10M]

2) Show that if \(\dfrac{1}{Q}\left[\dfrac{\partial p}{\partial y}-\dfrac{\partial Q}{\partial x}\right]\) is a function of \(x\) only, say, \(f ( x ),\) then \(F(x)=e^{\int f(x) d x}\) is an integrating factor of \(P dx + Q dy =0\)

[10M]

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

[10M]

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

[10M]

5) If \(\dfrac{d^{2} x}{d t^{2}}+\dfrac{g}{b}(x-a)=0\left( a , b \right.\) and \(g\) being positive constants) and \(x = a\), and \(\dfrac{d x}{d t}=0\) when \(t =0\), show that \(x=a+\left(a^{\prime}-a\right) \cos \sqrt{\dfrac{g}{b} t}\)

[10M]

6) Solve \(\left( D ^{2}-4 D +4\right) y =8 x ^{2} e ^{2 x } \sin 2 x\) where \(D \equiv \dfrac{d}{d x}\)

[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 \(\dfrac{x^{2}}{a^{2}+\lambda}+\dfrac{y^{2}}{b^{2}+\lambda}=1\) is self-orthogonal.

[10M]

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

[10M]

4) Solve \(y \dfrac{d^{2} y}{d x^{2}}-2\left(\dfrac{d y}{d x}\right)^{2}=y^{2}\).

[10M]

5) Solve \(\dfrac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=a \cos \omega t\) and discuss the nature of solution as \(\omega \xrightarrow{\text{hello}} \omega_{0}\).

[10M]

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

[10M]

1992

1) By eliminating the constants \(a\), \(b\) obtain the differential equation of which \(\mathrm{xy}=\mathrm{ae}^{\mathrm{x}}+\mathrm{be}^{-\mathrm{x}}+\mathrm{x}^{2}\) 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

\((4 x+3 y+1) d x+(3 x+2 y+1) d y=0\) 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:

\(\left(\mathrm{D}^{2}+4\right) y=\sin 2 \mathrm{x}\), given that when \(\mathrm{x}=0,\) then \(\mathrm{y}=0\) and \(\dfrac{d y}{d x}=2\)

[10M]

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

[10M]

8) Solve: \(\left(x^{2} D^{2}+x D-4\right) y=x^{2}\).

[10M]

1991

1) If the equation \(\mathrm{Mdx}+\mathrm{Ndy}=0\) is of the form \(\mathrm{f}_{1}\) (xy). \(\mathrm{ydx}+\mathrm{f}_{2}(\mathrm{xy}) . \mathrm{x} \mathrm{dy}=0,\) then show that \(\dfrac{1}{M x-N y}\).

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

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

4) Solve \(\dfrac{d^{4} y}{d x^{4}}-m^{4} y=\sin m x\).

5) Solve the differential equation

6) Solve the differential equation \(\dfrac{d^{2} y}{d x^{2}}-4 \dfrac{d y}{d x}-5 y=x e^{-x},\) given that \(y=0\) and \(\dfrac{d y}{d x}=0,\) when \(\mathrm{x}=0\).

1990

1.(a) If the equation \(\lambda^{n}+a_{1} \lambda^{n-1}+\ldots \ldots \ldots+a_{n}=0\) (in unknown \(\lambda\) ) has distinct roots \(\lambda_{1}, \lambda_{2}, \ldots \ldots \ldots \ldots \lambda_{n}\). Show that the constant coefficients of differential equation
\(\dfrac{d^{n} y}{d x^{n}}+a_{1} \dfrac{d^{n-1} y}{d x^{n-1}}+\ldots \ldots \ldots+a_{n-1} \dfrac{d y}{d x}+a_{n}=b\)

has the most general solution of the form \(y=c_{0}(x)+c_{1} e^{2, x}+c_{2} e^{2 x}+\ldots \ldots \ldots \ldots . .+c_{2} e^{\lambda x}\), where \(c_{1}, c_{2} \ldots \ldots . c_{n}\) 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 \(x^{2} \dfrac{d^{2} y}{d x^{2}}+2 x \dfrac{d y}{d x}+y=0\) is explicit form. If your answer contains imaginary quantities, recast it in a form free of those.

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

4) Verify that \(y=\left(\sin ^{-1} x\right)^{2}\) is a solution of \(\left(1-x^{2}\right)\dfrac{d^{2} y}{d x^{2}}-x \dfrac{d y}{d x}=2 .\) Find also the most general solution.

1989

1) Find the value of \(y\) which satisfies the equation \(\left(x y^{3}-y^{3}-x^{2} e^{x}\right)+3 x y^{2} \dfrac{d y}{d x}=0\); given that \(y=1\) when \(x=1\).

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

3) Solve the differential equation

\(\dfrac{d^{3} y}{d x^{3}}-\dfrac{d^{2} y}{d x^{2}}-6 \dfrac{d y}{d x}=1+x^{2}\).