2nd Order ODEs · Visual Guide

Constant coefficients?

General solution

$$y_{\text{general}} = y_{\text{homogeneous}} + y_{\text{particular}}$$

Yes

Characteristic eq.

Solve

$m^2 + p\,m + q = 0$

$\Delta = p^2 - 4q$

No

Reduction of order

We know $y_1$

$v(x)\!=\!\int \frac{e^{-\int P(x)\,dx}}{y_1^{2}}\,dx$

$y_2 = v\cdot y_1$

$y = C_1 y_1 + C_2 y_2$

Step 1

Compute $y_{\text{homogeneous}}$

Solve $y'' + P(x)y' + Q(x)y = 0$ → see left branch

Step 2

Compute $y_{\text{particular}}$

Based on discriminant $\Delta$

$R(x)$ simple and const. coeff.? Exp · Pol · Sin · Cos

Δ > 0

Distinct real roots · $m_1, m_2 \in \mathbb{R}$

$y(x) = C_1 e^{m_1 x} + C_2 e^{m_2 x}$

Δ = 0

Repeated root · $m_1 = m_2 = m$

$y(x) = e^{m x}\,(C_1 + C_2\,x)$

Δ < 0

Complex roots · $m_{1,2} = a \pm b\,i$

$y(x) = e^{a x}\bigl[C_1\cos(bx) + C_2\sin(bx)\bigr]$

Undetermined Coefficients

Requirements: coefficients $p$ and $q$ constant and $R(x)$ simple.
Propose a $y_p$ (trial solution) with the same form as $R(x)$, with coefficients to determine:

Exp $y_p = A\,e^{a x}$

Sin/Cos $y_p = A\sin(bx) + B\cos(bx)$

Pol $y_p = A x^{n} + \dots + E x + F$

If any term of the trial solution coincides with a term of $y_h$:

Multiply the entire trial solution by x

If it still coincides → multiply by x²

Ex: if $y_h$ contains $e^{2x}$ and the trial is $y_p = A e^{2x}$, use $y_p = A x\,e^{2x}$.

Differentiate $y_p$ and substitute into the original equation to determine the coefficients.

Variation of Parameters

1

Identify $y_1, y_2$ such that $y_h = C_1 y_1 + C_2 y_2$

2

Wronskian

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

3

Compute $v_1(x), v_2(x)$

$v_1(x) = -\!\int \tfrac{y_2(x)\,R(x)}{W(x)}\,dx$

$v_2(x) = \phantom{-}\!\int \tfrac{y_1(x)\,R(x)}{W(x)}\,dx$

4

Compute $y_p$

$y_p(x) = v_1(x)\, y_1(x) + v_2(x)\, y_2(x)$