2000

1) Show that \(3 \dfrac{d^{2} y}{d x^{2}}+4 x \dfrac{d y}{d x}-8 y=0\) has an integral which is a polynomial in \(x\). Deduce the general solution.

[10M]

2) Reduce \(\dfrac{d^{2} y}{d x^{2}}+P \dfrac{d y}{d x}+Q y=R,\) where, \(P, Q, R\) are functions of \(x\), to the normal form. Hence solve \(\dfrac{d^{2} y}{d x^{2}}-4 x \dfrac{d y}{d x}+\left(4 x^{2}-1\right) y=-3 e^{x^{2}} \sin 2 x\)

[10M]

3) Solve the differential equation \(y=x-2 a p+a p^{2} .\) Find the singular solution and interpret it geometrically.

[10M]

4) Show that \((4 x+3 y+1) d x+(3 x+2 y+1) d y=0\) represents a family of hyperbolas with a common axis and tangent at the vertex.

[10M]

5) Solve \(x \dfrac{d y}{d x}-y=(x-1)\left(\dfrac{d^{2} y}{d x^{2}}-x+1\right)\) by the method of variation of parameters.

[10M]

1999

1) Solve the differential equation \(\dfrac{xdx+ydy}{xdy-ydx}=(\dfrac{1-x^2-y^2}{x^2+y^2})^{1/2}\)

[10M]

2) Solve \(\dfrac{d^3y}{dx^3}-3\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}-2y=e^x+cos x\)

[10M]

3) By the method of variation of parameters, solve the differential equation \(\dfrac{d^2y}{dx^2}+a^2y=sec(ax)\).

[10M]

1998

1) Solve the differential equation: \(xy-\dfrac{dy}{dx}=y^3e^{-x^2}\)

[10M]

2) Show that the equation: \((4x+3y+1)dx+(3x+2y+1)dy=0\) represents a family of hyperbolas having as asymptotes the lines \(x+y=0,2x+y+1=0\).

[10M]

3) Solve the differential equation: \(y=3px+4p^2\)

[10M]

4) Solve the differential equation: \(\dfrac{d^2y}{dx^2}-5\dfrac{dy}{dx}+6y=e^{4x}(x^2+9)\)

[10M]

5) Solve the differential equation: \(\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+y=xsin x\)

[10M]

6) Solve the differential equation: \(x^3\dfrac{d^3y}{dx^3}+2x^2\dfrac{d^2y}{dx^2}+2y=10(x+\dfrac{1}{x})\).

[10M]

1997

1) Solve the initial value problem

[10M]

2) Solve \(\left(x^{2}-y^{2}+3 x-y\right) d x+\left(x^{2}-y^{2}+x-3 y\right) d y=0\).

[10M]

3) Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is \(3 \mathrm{~mm}\), and one hour later has been reduced to \(2 \mathrm{~mm}\). find an expression for the radius of the rain drop at any time.

[10M]

4) Solve \(\dfrac{d^{4} y}{d x^{4}}+6 \dfrac{d^{3} y}{d x^{3}}+11 \dfrac{d^{2} y}{d x^{2}} 6 \dfrac{d y}{d x}=20 e^{-2 x} \sin x\)

[10M]

5) Make use of the transformation \(y(x)=u(x) \sec x\) to obtain the solution of \(\mathrm{y}^{\prime \prime}-2 \mathrm{y}^{\prime} \tan \mathrm{x}+5 \mathrm{y}=0, \mathrm{y}^{\prime}(0)=0, \mathrm{y}^{\prime}(0)=\sqrt{6}\)
[10M]

6) Solve \(\begin{array}{c} (1+2 x)^{2} \dfrac{d^{2} y}{d x^{2}}-6(1+2 x) \dfrac{d y}{d x}+16 y=8(1+2 x)^{2} \\ y(0)=0, y^{\prime}(0)=2 \end{array}\)

[10M]

1996

1) Find the curves for which the sum of the reciprocals of the radius vector and polar subtangent is constant.

[8M]

8.(b) Solve: \(x^{2}(y-p x)=y p^{2}, p=\dfrac{d y}{d x}\)

[15M]

2) \(y \sin 2 x d x-\left(1+y^{2}+\cos ^{2} x\right) d y=0\)

[15M]

3) \(\dfrac{d^{2} y}{d x^{2}}+2 \dfrac{d y}{d x}+10 y+37 \sin 3 x=0\) Find the value of \(y\) when \(x=\pi / 2,\) if it is given that \(y=3\) and \(\dfrac{d y}{d x}=0,\) when \(x=0\)

[15M]

4) Solve: \(\dfrac{d^{4} y}{d x^{4}}+2 \dfrac{d^{3} y}{d x^{3}}-3 \dfrac{d^{2} y}{d x^{2}}=x^{2}+3 e^{2 x}+4 \sin x\)

[15M]

5) Solve: \(x^{3} \dfrac{d^{3} y}{d x^{3}}+3 x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}+y=x+\log x\)

[15M]

1995

1) Determine a family of curves for which the ratio of the y-intercept of the tangent to the radius vector is a constant.

[20M]

2) Solve \(\left(2 x^{2}+3 y^{2}-7\right) x d x-\left(3 x^{2}+2 y^{2}-8\right) y d y=0\)

[20M]

3) Test whether the equation \((x+y)^{2} d x-\left(y^{2}-2 x y-x^{2}\right) d y=0\) is exact and hence solve it.

[20M]

4) Solve \(x^{3} \dfrac{d^{3} y}{d x^{3}}+2 x^{2} \dfrac{d^{2} y}{d x^{2}}+2 y=10\left(x+\dfrac{1}{x}\right)\)

[20M]

5) Determine all real valued solutions of the equation \(y^{\prime \prime \prime} \cdot \mathrm{iy}^{\prime \prime}+\mathrm{y}^{\prime} \cdot \mathrm{iy}=0, y^{\prime}=\dfrac{d y}{d x}\)

[20M]

6) Find the solution of the equation \(y^{\prime \prime}+4 y=8 \cos 2 x\) give that \(y=0\) and \(y^{\prime}=2\) when \(x=0\)

[20M]