IFoS PYQs 5
2004
1) Determine the family of orthogonal trajectories of the family \(y=x+\operatorname{ce}^{-x}\).
[10M]
2) Show that the solution curve satisfying \(\left(x^{2}-x y\right) y^{4}=y^{3},\) where \(y \rightarrow 1\) as \(x \rightarrow 1, x \rightarrow \infty\) is a conic section. Identify the curve.
[10M]
3) Solve \((1+x)^{2} y^{\prime \prime}+(1+x) y^{\prime}+y=4 \cos (\ln (1+x)), y(0)=1, y(e-1)=\cos 1. 1\)
[10M]
4) Obtain the general solution of \(y^{\prime \prime}+2 y^{\prime}+2 y=4 e^{-x} x^{2} \sin x\)
[10M]
5) Find the general solution of \(\left(\mathrm{xy}^{3}+\mathrm{y}\right) \mathrm{dx}+2\left(\mathrm{x}^{2} \mathrm{y}^{2}+\mathrm{x}+\mathrm{y}^{4}\right) \mathrm{dy}=0\).
[10M]
6) Obtain the general solution of \(\left(\mathrm{D}^{4}+2 \mathrm{D}^{3}-2 \mathrm{D}\right) \mathrm{y}=\mathrm{x}+\mathrm{e}^{2x}\), where \(D y=\dfrac{d y}{d x}\).
[10M]
2003
1) Find the orthogonal trajectories of the family of eo-axial circles
\[x^{2}+y^{2}+2 g x+c=0\]where \(\mathrm{g}\) is a parameter.
[10M]
2) Find three solutions of
\[\dfrac{d^{3} y}{d x}-2 \dfrac{d^{2} y}{d x^{2}}-\dfrac{d y}{d x}-2 y=0\]which are linearly independent on every real interval.
[10M]
3) Solve and examine for singular solution:
\[y^{2}-2 p x y+p^{2}\left(x^{2}-1\right)=m^{2}\][10M]
4) Solve
\[x^{3} \dfrac{d^{3} y}{d x^{2}}+2 x^{2} \dfrac{d^{2} y}{d x^{2}}+2 y=10\left(x+\dfrac{1}{x}\right)\][10M]
5) Given \(y=x\) is one solutions of
\[\left(x^{3}+1\right) \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}+2 y=0\]find another linearly independent solution by reducing order and write the general solation.
[10M]
6) Solve by the method of variation of parameters
\(\dfrac{d^{2} y}{d x^{2}}+a^{2} y=\sec a x, a\) is real.
[10M]
2002
1) If \((D-a)^{4} e^{n x}\) is denoted by \(z\), prove that \(z \dfrac{\partial z}{\partial n}, \dfrac{\partial^{2} z}{\partial n^{2}}, \dfrac{\partial^{3} z}{\partial n^{3}}\) all vanish when \(n=a\). Hence show that \(e^{nx}, x e^{nx}, x^{2} e^{n x}, x^{3} e^{n x}\) are all solutions of \((D-a)^{4} y=0\). Here \(D\) stands for \(\dfrac{d}{d x}\).
[10M]
2) Solve \(4 x P^{2} \cdot(3 x+1)^{2}=0\) and examine for singular solutions and extraneous loci. Interpret the results geometrically.
[10M]
TBC
3)(i) Form the differential equation whose primitive is
\[y=A\left(\sin x+\dfrac{\cos x}{x}\right)+B\left(\cos x-\dfrac{\sin x}{x}\right)\][5M]
3)(ii) Prove that the orthogonal trajectory of system of parabolas belongs to the system itself.
[5M]
4) Using variation of parameters solve the differential equation
\(\dfrac{d^{2} y}{d x^{2}}-4 x \dfrac{d y}{d x}+\left(4 x^{2}-1\right) y=3 e^{4} \sin 2 x\)
[10M]
5.(i) Solve the equation by finding an integrating factor of
\((x+2) \sin y d x+x \cos y d y=0\).
[5M]
5.(ii) Verify that \(\phi(x)=x^{2}\) is a solution of \(y^{\prime \prime}- \dfrac{2}{x^{2}} y=0\) and find a second independent solution.
[5M]
6) Show that the solution of \(\left(D^{2 x+1}-1\right) y=0\), consists of \(\mathrm{Ae}^{x}\) and \(\mathrm{n}\) paris of terms of the form \(e^{\alpha x}(b_r \cos \alpha x+c_r \sin \alpha x)\), where \(a=\cos \dfrac{2 \pi r}{2 n+1}\) and \(\alpha=\sin \dfrac{2 \pi r}{2 n+1}\), \(r=1,2 \ldots \ldots, n\) and \(b_r\), \(c_r\) are arbitrary constants.
[10M]
2001
1) A tank of 100 liters capacity is initially full of water. Pure water is allowed to nun into the tank at the rate of 1 liter per minute and at the same time salt water containing \(\dfrac{1}{4} \mathrm{kg}\) of salt per liter flows into the tank at the rate of 1 liter per minute. The mixture (there is perfect mixing in the tank at all times) flows out at the rate of 2 liters per minute. Form the differential equation and find the amount of salt in the tank after \(t\) minutes. Find this when \(t=50\) minutes.
[10M]
2) A constant coefficient differential equation has auxiliary eqution expressible in factored form as \(P(m)=m^{3}(m-1)^{2}\left(m^{2}+2 m+5\right)^{2} .\) What is the order of the differential equation and find it general solution.
[10M]
3) Solve \(x^{2}\left(\dfrac{d y}{d x}\right)^{2}+y(2 x+y) \dfrac{d y}{d x}+y^{2}=0\).
[10M]
4) Using differential equations, show that the system of confocal conics given by
\[\dfrac{x^{2}}{a^{2}+\lambda}+\dfrac{y^{2}}{b^{2}+\lambda}=1,\]\(\lambda\) real is self-orthogonal.
[10M]
5) Solve \(\left(1-x^{2}\right) \dfrac{d^{2} y}{d x^{2}}-x \dfrac{d y}{d x}-a^{2} y=0\) given that \(y=e^{a \sin^{-1} x\) is one solution of this equation.
[10M]
6) Find a general solution of \(y^{\prime \prime}+y=\tan x, \dfrac{-\pi}{2}<x<\dfrac{\pi}{2}\) by variation of parameters.
[10M]
2000
1) Solve \(\left(x^{2}+y^{2}\right)(1+P)^{2}-2(x+y)(1+p)(x+y p)+(x+y p)^{2}=0\) \(P=\dfrac{d y}{d x} .\) Interpret geometrically the factors in the \(\mathrm{P}\). and C-discriminants of the equation \(8 p^{3} x=y\left(12 p^{2}-9\right)\).
[20M]
2) Solve
(i) \(\dfrac{d^{2} y}{d x^{2}}+\dfrac{2}{x} \dfrac{d y}{d x}+\dfrac{a^{2}}{x^{4}} y=0\)
(ii) \(\dfrac{d^{2} y}{d x^{2}}+(\tan x-3 \cos x) \dfrac{d y}{d x}+2 y \cos ^{2} x=\cos ^{4} x\) by varying parameters.
[20M]