Variable Coefficients ODEs
We will cover following topics
ODEs With Variable Coefficients
Equations of the form \(\dfrac{d^{2} y}{d x^{2}}+p(x) \dfrac{d y}{d x}+q(x) y=r(x)\) are known as second order linear equations with variable coefficients.
Euler-Cauchy Equation
An equation of the form \(a_{n} x^{n} y^{(n)}(x)\)+\(a_{n-1} x^{n-1} y^{(n-1)}(x)\)+\(\cdots\)+\(a_{0} y(x)=0\) is known as Euler-Cauchy equation.
The equation is converted into an equation with constant coefficients by first substituting \(x=e^u\).
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:
\[y=complementary \text{ }function + particular \text{ } integral\]Determination Of Complete Solution When One Solution Is Known
If a particular solution \(y_1(x)\) of the homogeneous linear second order equation is known, the original equation can be converted to a linear first order equation using the substitution \(y=y_1(x)z(x)\) 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 \(y_1(x)\) must also be known.
PYQs
ODEs with Variable Coefficients
1) Solve the differential equation
\[\dfrac{d^2y}{dx^2}+(3sin x-cot x)\dfrac{dy}{dx}+2ysin^2x=e^{-cos x}sin^2x\][2019, 10M]
2) Solve the differential equation: \(x \dfrac{d^{2} y}{d x^{2}}-\dfrac{d y}{d x}-4 x^{3} y=8 x^{3} \sin \left(x^{2}\right)\)
[2017, 8M]
3) Solve the following differential equation: \(x \dfrac{d^{2} y}{d x^{2}}-2(x+1) \dfrac{d y}{d x}+(x+2) y=(x-2) e^{2 x}\), when \(e^{x}\) is a solution to its corresponding homogeneous differential equation.
[2014, 15M]
4) Solve the ordinary differential equation \(x(x-1) y^{\prime \prime}-(2 x-1) y^{\prime}+2 y=x^{2}(2 x-3)\).
[2012, 20M]
5) Solve: \((x+2) \dfrac{d^{2} y}{d x^{2}}-(2 x+5) \dfrac{d y}{d x}+2 y=(x+1) e^{x}\).
[2004, 15M]
6) Solve the folllowing differential equation:
\[(1-x^2)\dfrac{d^2y}{dx^2} -4x \dfrac{dy}{dx}-(1+x^2)y=x\][2004, 15M]
7) Solve \(\left(1+x\right)^{2} y^{\prime \prime}+(1+x) y^{\prime}+y=\sin 2[\log (1+x)]\).
[2003, 15M]
Euler-Cauchy Equation
1) Obtain the singular solution of the differential equation
\[(\dfrac{dy}{dx})^2(\dfrac{y}{x})^2cot^2\alpha-2(\dfrac{dy}{dx})(\dfrac{y}{x})+(\dfrac{dy}{dx})^2cosec^2\alpha=1\]Also 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 \(x^2y''-2xy'+2y=x^3 \sin x\) 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
\[x^2y^{\prime \prime} -4xy{\prime}+6y=x^4 \sec^2 x\]by variation of parameters.
[2003, 15M]
10) Using the method of variation of parameteres, find the solution of
\[\dfrac{d^2y}{dx^2} -2 \dfrac{dy}{dx}+y=x e^x \sin x\]with \(y(0)=0\) 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]