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″−2xy′+2y=x3sinx 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 x4d4ydx4+6x3d3ydx3+4x2d2ydx2−2xdydx−4y=x2+2cos(logex).
[2015, 13M]
5) Solve the differential equation: x3d3ydx3+3x2d2ydx2+xdydx+8y=65cos(logex).
[2014, 20M]
6) Solve the differential equation x3y′′−3x2y′+xy=sin(lnx)+1.
[2008,15M]
7) Solve the equation 2x2d2ydx2+3xdydx−3y=x3.
[2007, 15M]
8) Solve the differential equation x2d3ydx3+2xd2ydx2+2yx=10(1+1x2).
[2006, 15M]
9) Solve the differential equation: [(x+1)4D3+2(x+1)3D2−(x+1)2D+(x+1)]y=1(x+1).
[2005, 15M]
10) Find the values of λ for which all solutions of x2d2ydx2+3xdydx−λy=0 tend to zero as x→∞.
[2002, 12M]
11) Solve: x2d2ydx2−xdydx−3y=x2logex.
[2001, 12M]
Determination of Complete Solution when One Solution is known
1) Using the method of variation of parameters, solve the differential equation (D2+2D+1)y=e−xlog(x),[D≡ddx].
[2016, 15M]
2) Using the method of variation of parameters, solve the differential equation d2ydx2+a2y=secax.
[2013, 15M]
3) Using the method of variation of parameters, solve the second order differential equation d2ydx2+4y=tan2x.
[2011, 15M]
4) Use the method of variation of parameters to find the general solution of x2y′′−4xy′+6y=−x4sinx.
[2008, 12M]
5) Use the method of variation of parameters to find the general solution of the equation d2ydx2+3dydx+2y=2ex.
[2007, 15M]
6) Solve the differential equation (D2−2D+2)y=extanx,D≡dydx by the method of variation of parameters.
[2006, 15M]
7) Solve the differential equation (sinx−xcosx)y′′−xsinxy′+ysinx=0 given that y=sinx is a solution of this equation.
[2005, 15M]
8) Solve the differential equation x2y′′−2xy′+2y=xlogx,x>0 by variation of parameters.
[2005, 15M]
9) Solve the differential equation
x2y′′−4xy′+6y=x4sec2xby variation of parameters.
[2003, 15M]
10) Using the method of variation of parameteres, find the solution of
d2ydx2−2dydx+y=xexsinxwith y(0)=0 and (dydx)x=0=0
[2002, 15M]
11) Using the method of variation of parameters, solve:
d2ydx2+4y=4tan2x[2001, 15M]