Variable Coefficients ODEs
We will cover following topics
ODEs With Variable Coefficients
Equations of the form d2ydx2+p(x)dydx+q(x)y=r(x) are known as second order linear equations with variable coefficients.
Euler-Cauchy Equation
An equation of the form anxny(n)(x)+an−1xn−1y(n−1)(x)+⋯+a0y(x)=0 is known as Euler-Cauchy equation.
The equation is converted into an equation with constant coefficients by first substituting x=eu.
The equation can then be solved by first finding the complementary function using the homogeneous equation and then finding the particular integral.
The complete solution is written as:
y=complementary function+particular integralDetermination Of Complete Solution When One Solution Is Known
If a particular solution y1(x) of the homogeneous linear second order equation is known, the original equation can be converted to a linear first order equation using the substitution y=y1(x)z(x) and the subsequent replacement z′(x)=u.
Another way to reduce the order is based on the Liouville formula. In this case, a particular solution y1(x) must also be known.
PYQs
ODEs with Variable Coefficients
1) Solve the differential equation
d2ydx2+(3sinx−cotx)dydx+2ysin2x=e−cosxsin2x[2019, 10M]
2) Solve the differential equation: xd2ydx2−dydx−4x3y=8x3sin(x2)
[2017, 8M]
3) Solve the following differential equation: xd2ydx2−2(x+1)dydx+(x+2)y=(x−2)e2x, when ex is a solution to its corresponding homogeneous differential equation.
[2014, 15M]
4) Solve the ordinary differential equation x(x−1)y′′−(2x−1)y′+2y=x2(2x−3).
[2012, 20M]
5) Solve: (x+2)d2ydx2−(2x+5)dydx+2y=(x+1)ex.
[2004, 15M]
6) Solve the folllowing differential equation:
(1−x2)d2ydx2−4xdydx−(1+x2)y=x[2004, 15M]
7) Solve (1+x)2y′′+(1+x)y′+y=sin2[log(1+x)].
[2003, 15M]
Euler-Cauchy Equation
1) Obtain the singular solution of the differential equation
(dydx)2(yx)2cot2α−2(dydx)(yx)+(dydx)2cosec2α=1Also find the complete primitive of the given differential equation. Give the geometrical interpretations of the complete primitive and singular solution.
[2019, 15M]
2) Find the linearly independent solutions of the corresponding homogeneous differential equation of the equation x2y″ and then find the general solution of the given equation by the method of variation of parameters.
[2019, 15M]
3) Solve:
(1+x)^2y''+(1+x)y'+y=4cos(log(1+x))[2018, 13M]
4) Solve x^{4} \dfrac{d^{4} y}{d x^{4}}+6 x^{3} \dfrac{d^{3} y}{d x^{3}}+4 x^{2} \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}-4 y=x^{2}+2 \cos \left(\log _{e} x\right).
[2015, 13M]
5) Solve the differential equation: 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}+8 y=65 \cos \left(\log _{e} x\right).
[2014, 20M]
6) Solve the differential equation x^{3} y^{\prime \prime}-3 x^{2} y^{\prime}+x y=\sin (\ln x)+1.
[2008,15M]
7) Solve the equation 2 x^{2} \dfrac{d^{2} y}{d x^{2}}+3 x \dfrac{d y}{d x}-3 y=x^{3}.
[2007, 15M]
8) Solve the differential equation x^{2} \dfrac{d^{3} y}{d x^{3}}+2 x \dfrac{d^{2} y}{d x^{2}}+2 \dfrac{y}{x}=10\left(1+\dfrac{1}{x^{2}}\right).
[2006, 15M]
9) Solve the differential equation: \left[(x+1)^{4} D^{3}+2(x+1)^{3} D^{2}-(x+1)^{2} D+(x+1)\right] y=\dfrac{1}{(x+1)}.
[2005, 15M]
10) Find the values of \lambda for which all solutions of x^{2} \dfrac{d^{2} y}{d x^{2}}+3 x \dfrac{d y}{d x}-\lambda y=0 tend to zero as x \rightarrow \infty.
[2002, 12M]
11) Solve: x^{2} \dfrac{d^{2} y}{d x^{2}}-x \dfrac{d y}{d x}-3 y=x^{2} \log _{e} x.
[2001, 12M]
Determination of Complete Solution when One Solution is known
1) Using the method of variation of parameters, solve the differential equation \left(D^{2}+2 D+1\right) y=e^{-x} \log (x),\left[D \equiv \dfrac{d}{d x}\right].
[2016, 15M]
2) Using the method of variation of parameters, solve the differential equation \dfrac{d^{2} y}{d x^{2}}+a^{2} y=\sec a x.
[2013, 15M]
3) Using the method of variation of parameters, solve the second order differential equation \dfrac{d^{2} y}{d x^{2}}+4 y=\tan 2 x.
[2011, 15M]
4) Use the method of variation of parameters to find the general solution of x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=-x^{4} \sin x.
[2008, 12M]
5) Use the method of variation of parameters to find the general solution of the equation \dfrac{d^{2} y}{d x^{2}}+3 \dfrac{d y}{d x}+2 y=2 e^{x}.
[2007, 15M]
6) Solve the differential equation \left(D^{2}-2 D+2\right) y=e^{x} \tan x, D \equiv \dfrac{d y}{d x} by the method of variation of parameters.
[2006, 15M]
7) Solve the differential equation (\sin x-x \cos x) y^{\prime \prime}-x \sin x y^{\prime}+y \sin x=0 given that y=\sin x is a solution of this equation.
[2005, 15M]
8) Solve the differential equation x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x \log x, x>0 by variation of parameters.
[2005, 15M]
9) Solve the differential equation
x^2y^{\prime \prime} -4xy{\prime}+6y=x^4 \sec^2 xby variation of parameters.
[2003, 15M]
10) Using the method of variation of parameteres, find the solution of
\dfrac{d^2y}{dx^2} -2 \dfrac{dy}{dx}+y=x e^x \sin xwith y(0)=0 and \left( \dfrac{dy}{dx} \right)_{x=0}=0
[2002, 15M]
11) Using the method of variation of parameters, solve:
\dfrac{d^2y}{dx^2} +4y=4 \tan 2x[2001, 15M]