We will cover following topics

ODEs with Constant Coefficients
Complementary Function
Particular Integral
- Method of Undetermined Coefficients
- Method of Variation of Parameters
General and Particular Solutions
- General Solution
- Particular Solution:
PYQs
- ODEs with Constant Coefficients

ODEs with Constant Coefficients

Equations of the form $a \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = f (x)$ 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_{C F} + y_{P I}

To determine CF, we first form the homogenous equation and then convert it into the auxiliary equation. We, then find the CF.
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 \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = 0

Auxiliary Equation: The auxiliary equation of $a \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = 0$ is given by $a k^{2} + b k + c = 0, where y = e^{k x}$ .

Now, the three cases arise:

Case I: If $b^{2} > 4 a c$ , then the auxiliary equation has real and distinct roots $k_{1}$ and $k_{2}$ , and the complementary function is given by:

y_{C F} (x) = A e^{k_{1} x} + B e^{k_{2} x}

Case II: If $b^{2} = 4 a c$ , then the auxiliary equation has real and equal roots $k$ , and the complimentary function is given by:

y_{C F} = (A + B x) e^{k x}

Case III: If $b^{2} < 4 a c$ , then the auxiliary equation has complex roots $α + β i$ and $α - β i$ , and the complementary function is given by:

y_{C F} = e^{α x} (A \cos β x + B \sin β x)

Quickly find CF

To summarise, for the given ODE $a \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = 0$ , the complimentary function can be found using below conditions:

Condition	Complimentary Function
$b^{2} > 4 a c$	$y_{C F} (x) = A e^{k_{1} x} + B e^{k_{2} x}$
$b^{2} = 4 a c$	$y_{C F} = (A + B x) e^{k x}$
$b^{2} < 4 a c$	$y_{C F} = e^{α x} (A \cos β x + B \sin β x)$

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 | |:———-: |:————-:| | $k$ | $C$ | | $k x$ | $C x + D$ | | $k x^{2}$ | $C x^{2} + D x + E$ | | $k \sin x$ or $k \cos x$ | $C \cos x$ + $D \sin x$ | | $k \sin h x$ or $k \cos h x$ | $C \cos h x$ + $D \sin h x$ | | $e^{k x}$ | $C e^{k x}$ | | $e^{r x}$ (r: root of AE) | $C x e^{r x}$ or $C x^{2} e^{r x} | |$ 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:

$y_{p} (x)$ = $- y_{1} (x) \int \frac{y_{2} (x) f (x)}{W (y_{1}, y_{2})} d x$ + $y_{2} (x) \int \frac{y_{1} (x) f (x)}{W (y_{1}, y_{2})} d x$ ,

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:

$W (y_{1}, y_{2}) = | \begin{matrix} y_{1} & y_{2} \\ y_{1}^{'} & y_{2}^{'} \end{matrix} |$

General and Particular Solutions

General Solution

The general solution is the sum of the complementary function and the particular integral.

y (x) = y_{C F} (x) + y_{P I} (x)

Particular Solution:

In the particular solution, the unknown coefficients in the general solution are found by imposing the given boundary conditions on the general solution.

PYQs

ODEs with Constant Coefficients

1) Solve:

y^{″} + 16 y = 32 s e c 2 x

[2018, 13M]

2) Solve:

y^{″} - y = x^{2} e^{2 x}

[2018, 10M]

3) Solve:

y^{‴} - 6 y^{″} + 12 y^{'} - 8 y = 12 e^{2 x} + 27 e^{- x}

[2018, 10M]

4) Solve the following initial value differential equations: $20 y^{''} + 4 y^{'} + y = 0$ , $y (0) = 3.2$ , $y^{'} (0) = 0$ .

[2017, 7M]

5) Find a particular integral of $\frac{d^{2} y}{d x^{2}} + y = e^{x / 2} \sin \frac{x \sqrt{3}}{2}$ .

[2016, 10M]

6) Find the general solution of the equation $x^{2} \frac{d^{3} y}{d x^{3}} - 4 x \frac{d^{2} y}{d x^{2}} + 6 \frac{d y}{d x} = 4$ .

[2016, 15M]

7) Find the general solution of the equation $x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + y = \ln x \sin (\ln x)$ .

[2013, 15M]

8) Find the general solution of the equation $y^{'''} - y^{''} = 12 x^{2} + 6 x$ .

[2012, 20M]

9) Obtain the general solution of the second order ordinary differential equation $y^{''} - 2 y^{'} + 2 y = x + e^{x} \cos x$ , where dashes denote derivatives w.r.t. $x$ .

[2011, 15M]

10) Use the method of undetermined coefficients to find the particular solutions of $y^{''} + y = \sin x + (1 + x^{2}) e^{x}$ and hence find its general solution.

[2010, 20M]

11) Find the Wronskian of the set of functions: ${3 x^{3}, | 3 x^{3} |}$ on the interval $[- 1, 1]$ and determine whether the set is linearly dependent on $[- 1, 1]$ .

[2009, 12M]

12) Obtain the general solution of $[D^{3} - 6 D^{2} + 12 D - 8] y = 12 (e^{2 x} + \frac{9}{4} e^{- x})$ , where $D \equiv \frac{d y}{d x}$ .

[2007, 15M]

13) Solve: $(D^{4} - 4 D^{2} - 5) y = e^{x} (x + \cos x)$ .

[2004, 15M]

14) Solve $(D^{5} - D) = 4 (e^{x} + \cos x + x^{3})$ , where $D \equiv \frac{d y}{d x}$ .

[2003, 15M]

15) Solve: $(D - 1) (D^{2} - 2 D + 2) y = e^{x}$ , where $D \equiv \frac{d y}{d x}$ .

[2002, 15M]

16) Solve: ${(D^{2} + 1)}^{2} y = 24 x \cos x$ 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]

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