Constant Coefficients ODEs
We will cover following topics
ODEs with Constant Coefficients
Equations of the form \(a \dfrac{d^{2} y}{d x^{2}}+b \dfrac{d y}{d x}+c y=f(x)\) are known as second order linear equations with constant coefficients.
To solve an ODE with constant coefficients, we need to determine the complimentary function (CF) and particular integral (PI).
The solution can then then be written as:
\[y = y_{CF} + y_{PI}\]-
To determine CF, we first form the homogenous equation and then convert it into the auxiliary equation. We, then find the CF.
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PI can be obtained either by method of undetermined coefficients or by the method of vatiation of parameters.
The detailed steps are illustrated below.
Complementary Function
Homogeneous Equation: The homogeneous equation is derived by simply replacing \(f(x)\) by zero in the ODE:
\[a \dfrac{d^{2} y}{d x^{2}}+b \dfrac{d y}{d x}+c y=0\]Auxiliary Equation: The auxiliary equation of \(a\dfrac{d^{2} y}{d x^{2}}+b \dfrac{d y}{d x}+c y=0\) is given by \(a k^{2}+b k+c=0 \text {, where } y=e^{k x}\).
Now, the three cases arise:
Case I: If \(b^2>4ac\), then the auxiliary equation has real and distinct roots \(k_1\) and \(k_2\), and the complementary function is given by:
\[y_{CF}(x)=A e^{k_{1} x}+B e^{k_{2} x}\]Case II: If \(b^2=4ac\), then the auxiliary equation has real and equal roots \(k\), and the complimentary function is given by:
\[y_{CF}=(A+B x) e^{k x}\]Case III: If \(b^2< 4ac\), then the auxiliary equation has complex roots \(\alpha+\beta i\) and \(\alpha-\beta i\), and the complementary function is given by:
\[y_{CF}=e^{\alpha x}(A \cos \beta x+B \sin \beta x)\]Quickly find CF
To summarise, for the given ODE \(a\dfrac{d^{2} y}{d x^{2}}+b \dfrac{d y}{d x}+c y=0\), the complimentary function can be found using below conditions:
| Condition | Complimentary Function |
|---|---|
| \(b^2>4ac\) | \(y_{CF}(x)=A e^{k_{1} x}+B e^{k_{2} x}\) |
| \(b^2=4ac\) | \(y_{CF}=(A+B x) e^{k x}\) |
| \(b^2<4ac\) | \(y_{CF}=e^{\alpha x}(A \cos \beta x+B \sin \beta x)\) |
Particular Integral
The particular integral of the and can be calculated using either of the below methods:
- Method of undetermined coefficients
- Method of variation of parameters
Method of Undetermined Coefficients
This method involves making an educated guess for the particular integral. The following table can be used for most of the problems:
| f(x) | Particular Integral | |:———-: |:————-:| | \(k\) | \(C\) | | \(kx\) | \(Cx+D\) | | \(kx^2\) | \(Cx^2+Dx+E\) | | \(k \sin x\) or \(k \cos x\) | \(C \cos x\) + \(D \sin x\) | | \(k \sin hx\) or \(k \cos hx\) | \(C \cos hx\) + \(D \sin hx\) | | \(e^{kx}\) | \(Ce^{kx}\) | | \(e^{rx}\) (r: root of AE) | \(Cxe^{rx}\) or \(Cx^2e^{rx} | |\)kx\(|\)Cx+D\(| |\)kx\(|\)Cx^2+Dx+E$$ |
Method of Variation of Parameters
This method is more general and can be used for any function \(f(x)\). Using the method of variation of parameters, the particular integral can be calculated as below:
\(y_{p}(x)\)=\(-y_{1}(x) \int \dfrac{y_{2}(x) f(x)}{W\left(y_{1}, y_{2}\right)} d x\)+\(y_{2}(x) \int \dfrac{y_{1}(x) f(x)}{W\left(y_{1}, y_{2}\right)} d x\),
where \(y_1(x)\) and \(y_2(x)\) are the solutions of the homogeneous equation,
Also, \(W(y_1,y_2)\) is known as the Wronskian and is calculated as:
\(W\left(y_{1}, y_{2}\right)= \begin{vmatrix} y_1 & y_2 \\ y_1^{\prime} & y_2^{\prime} \end{vmatrix}\)
General and Particular Solutions
General Solution
The general solution is the sum of the complementary function and the particular integral.
\[y(x) = y_{CF}(x) + y_{PI}(x)\]PYQs
ODEs with Constant Coefficients
1) Solve:
\[y''+16y=32sec 2x\][2018, 13M]
2) Solve:
\[y''-y=x^2e^{2x}\][2018, 10M]
3) Solve:
\[y'''-6y''+12y'-8y=12e^{2x}+27e^{-x}\][2018, 10M]
4) Solve the following initial value differential equations: \(20 y^{\prime \prime}+4 y^{\prime}+y=0\), \(y(0)=3.2\), \(y^{\prime}(0)=0\).
[2017, 7M]
5) Find a particular integral of \(\dfrac{d^{2} y}{d x^{2}}+y=e^{x / 2} \sin \dfrac{x \sqrt{3}}{2}\).
[2016, 10M]
6) Find the general solution of the equation \(x^{2} \dfrac{d^{3} y}{d x^{3}}-4 x \dfrac{d^{2} y}{d x^{2}}+6 \dfrac{d y}{d x}=4\).
[2016, 15M]
7) Find the general solution of the equation \(x^{2} \dfrac{d^{2} y}{d x^{2}}+x \dfrac{d y}{d x}+y=\ln x \sin (\ln x)\).
[2013, 15M]
8) Find the general solution of the equation \(y^{\prime \prime \prime}-y^{\prime \prime}=12 x^{2}+6 x\).
[2012, 20M]
9) Obtain the general solution of the second order ordinary differential equation \(y^{\prime \prime}-2 y^{\prime}+2 y=x+e^{x} \cos x\), where dashes denote derivatives w.r.t. \(x\).
[2011, 15M]
10) Use the method of undetermined coefficients to find the particular solutions of \(y^{\prime \prime}+y=\sin x+\left(1+x^{2}\right) e^{x}\) and hence find its general solution.
[2010, 20M]
11) Find the Wronskian of the set of functions: \(\left\{3 x^{3},\left \vert 3 x^{3}\right \vert \right\}\) on the interval \([-1,1]\) and determine whether the set is linearly dependent on \([-1,1]\).
[2009, 12M]
12) Obtain the general solution of \(\left[D^{3}-6 D^{2}+12 D-8\right] y=12\left(e^{2 x}+\dfrac{9}{4} e^{-x}\right)\), where \(D \equiv \dfrac{d y}{d x}\).
[2007, 15M]
13) Solve: \(\left(D^{4}-4 D^{2}-5\right) y=e^{x}(x+\cos x)\).
[2004, 15M]
14) Solve \(\left(D^{5}-D\right)=4\left(e^{x}+\cos x+x^{3}\right)\), where \(D \equiv \dfrac{d y}{d x}\).
[2003, 15M]
15) Solve: \((D-1)\left(D^{2}-2 D+2\right) y=e^{x}\), where \(D \equiv \dfrac{d y}{d x}\).
[2002, 15M]
16) Solve: \(\left(D^{2}+1\right)^{2} y=24 x \cos x\) given that \(y=D y=D^{2} y=0\) and \(D^{3} y=12\) when \(x=0\).
[2001, 15M]
17) Find the general solution of \(a y p^{2}+(2 x-b) p-y=0\), \(a>0\).
[2001, 15M]