IFoS PYQs 4
2008
1) Show that the functions \(y_{1}(x)=x^{2}\) and \(y_{2}(x)=x^{2} \log _{e} x\) are linearly independent. Obtain the differential equation that has \(y_{1}(x)\) and \(y_{2}(x)\) as the independent solutions.
[10M]
2) Solve the following ordinary differential equatjon of the second degree:
\(y\left(\dfrac{d y}{d x}\right)^{2}+(2 x-3) \dfrac{d y}{d x}-y=0\)
[10M]
3) Reduce the equation \(\left(x \dfrac{d y}{d x}-y\right)\left(x-y \dfrac{d y}{d x}\right)=2 \dfrac{d y}{d x}\) to Clairaut’s form and obtain thereby the singular integral of the above equation.
[10M]
4) Solve:
\((1+x)^{2} \dfrac{d^{2} y}{d x^{2}}+(1+x) \dfrac{d y}{d x}+y=4 \cos \log _{e}(1+x)\)
[10M]
5) Find the general solution of the equation
\(\dfrac{d^{2} y}{d x^{2}}-\cot x \dfrac{d y}{d x}-(1-\cot x) y=e^{x} \sin x\)
[10M]
6) Use the method of variation of parameters to solve the differential equation
\(x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}-y=x^{2} e^{x}\).
[10M]
2007
1) Find the orthogonal trajectories of the family of the curves \(\dfrac{x^{2}}{a^{7}}+\dfrac{y^{2}}{b^{2}+\lambda}=1\), \(\lambda\) being a parameter.
[10M]
2) Show that \(e^{2 x}\) and \(e^{3 x}\) are linearly independent solutions of
\[\dfrac{d^{2} y}{d x^{2}}-5 \dfrac{d y}{d x}+6 y=0\]Find the general solution when \(y(0)=0\) and \(\dfrac{d y}{d x}=1\) at \(x=0\).
[10M]
3) Find the family of curves whose tangents form an \(\dfrac{\pi}{4}\) angle with the byperbola \(xy=C\).
[10M]
4) Apply. the method of variation of parameters to solve
\[\left(D^{2}+a^{2}\right) y= cosec ax\][10M]
5) Solve
\[\dfrac{d^{2} y}{d x^{2}}+\dfrac{2}{x} \dfrac{d y}{d x}+\dfrac{a^{2}}{x^{4}} y=0\]By using the method of removal of first derivate.
[10M]
6) Find the general solution of
\(\left(1-x^{2}\right) \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}+3 y=0,\) if \(y=x\) is a solution of it
[10M]
2006
1) From \(x^{2}+y^{2}+2 a x+2 b y+c=0,\) derive differential equation not containing \(a, b\) or \(c\).
[10M]
2) Discuss the solution of the differential equation \(y^{2}\left[1+\left(\dfrac{d y}{d x}\right)^{2}\right]=a^{2}\).
[10M]
3) Solve
\(x \dfrac{d^{2} y}{d x^{2}}+(1-x) \dfrac{d y}{d x}-y=e^{x}\)
[10M]
4) Solve
\(\dfrac{d^{4} y}{d x^{4}}-y=x \sin x\)
[10M]
5) Solve
\(x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}-y=x^{2} e^{x}\)
[10M]
6) Reduce
\(x y\left(\dfrac{d y}{d x}\right)^{2}-\left(x^{2}+y^{2}+1\right) \dfrac{d y}{d x}+x y=0\)
to Clairaut’s form and find its singular solution.
[10M]
2005
1) Form the differential equation that represents all parabolas each of which has latus rectum \(4 \mathrm{a}\) and whose axes are parallel to the \(x-axis\).
[10M]
2.(i) The auxiliary polynomial of a certain homogeneous linear differential equation with constant coefficients in factored form is \(P(m)=m^{4}(m-2)^{6}\left(m^{2}-6 m+25\right)^{3}\) What is the order of the differential equation and write a general solution?
[5M]
2.(ii) Find the equation of the one parameter family of parabolas given by \(y^{2}=2 c x+c^{2}\), \(c\) real and show that this family is self-orthogonal.
[5M]
3) Solve and examine for singular solution the following equation.
\(p^{2}\left(x^{2}-a^{2}\right)-2 p x y+y^{2}-b^{2}=0\).
[10M]
4) Solve the differential equation \(\dfrac{d^{2} y}{d x^{2}}+9 y=\sec 3 x\).
[10M]
5) Given \(y=x+\dfrac{1}{x}\) is one solution, solve the differential equation \(x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}-y=0\) by reduction of order method.
[10M]
6) Find the general solution of the differential equation \(\dfrac{d^{2} y}{d x^{2}}-2 y \dfrac{d y}{d x}-3 y=2 e^{x}-10 \sin x\) by the method of undetermined coefficients.
[10M]