06 2nd Order DE

Last Updated: 2 years ago2023-01-26 ; Contributors: AhmedThahir

Types

Complete Equation

$$y'' + P(x) y' + Q(x) y = R(x)$$

Also called non-homogeneous DE Particular Solution of complete equation: $y_p(x)$ If $y(x)$ is the solution, then it is given by

$$y(x) = y_g + y_p$$

Reduced Equation

Complete equation with $R(x) = 0$

$$y'' + P(x) y' + Q(x) y = 0$$

Also called as homogeneous DE

$$y(x) = y_g \quad (y_p(x) = 0)$$

Theorems

1

If $y_1(x)$ and $y_2(x)$ are 2 solutions of reduced DE, then $\set{c_1 y_1(x) + c_2 y_2(x)}$ is another solution of the reduced DE for any constants $c_1, c_2$

2

If $y_1(x)$ and $y_2(x)$ are 2 solutions of reduced DE, then they are linearly-dependent $\iff$ their wronskian = 0

$$W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = 0$$

Else, they are linearly-independent

eg:

• $y_1 = x^2, y_2 = \frac{3}{2} x^2$ - linear dependent
• $y_1 = x, y_2 = x^2$ - linearly independent

3

If $y_1(x)$ and $y_2(x)$ are 2 linearly-independent solutions of reduced DE, then $y(x) = c_1 y_1(x) + c_2 y_2(x)$ is called general solution

Solving

1. Sub $y = y_1(x)$ and $y = y_2(x)$ in the given equation
2. Show that LHS = RHS

Principle of Superposition

If the given DE is of the form

\[ y'' + py' + qy = f(x) + g(x) \]

Solution is given by

\[ \begin{aligned} y'' + py' + qy &= 0 &\to y_g \\ y'' + py' + qy &= f(x) &\to y_{p_1} \\ y'' + py' + qy &= g(x) &\to y_{p_2} \\ \implies y &= y_g + y_{p_1} + y_{p_2} \end{aligned} \]

This superposition of the solutions is called as principle of superposition.

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