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06 2nd Order DE

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.

Last Updated: 2023-01-25 ; Contributors: AhmedThahir

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