Second-Order Linear ODE Solver | Homogeneous & Non-Homogeneous

SECOND-ORDER LINEAR ODE SOLVER

Homogeneous & Non-Homogeneous ($ay'' + by' + cy = g(x)$)

Differential Equations

Select Differential Equation Scenario: Real Distinct Roots: y'' - 5y' + 6y = 0 Repeated Roots: y'' - 4y' + 4y = 0 Complex Conjugate Roots: y'' + 2y' + 5y = 0 Undetermined Coeffs (Poly): y'' + y' - 2y = 4x Harmonic Oscillator: y'' + 4y = cos(x)

Coefficient $a$ ($y''$):

Coefficient $b$ ($y'$):

Coefficient $c$ ($y$):

Forcing Function $g(x)$: 0 (Homogeneous) Polynom. (Ax + B) Exponential (e^{kx}) Trigonometric (cos(kx))

Initial $x_0$ (Optional):

Initial $y(x_0)$ (Optional):

Initial Derivative $y'(x_0)$ (Optional):

Analytical Solution Breakdown

Characteristic Roots ($\r_1, r_2$) r = 2, 3

Discriminant ($\Delta = b^2 - 4ac$) 1.0000

General Solution $y(x) = y_h(x) + y_p(x)$ y(x) = C1*e^(2x) + C2*e^(3x)

Step-by-Step Symbolic Derivation

Second-Order Linear Ordinary Differential Equations

A second-order linear ordinary differential equation with constant coefficients takes the general canonical form:

$$a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = g(x)$$

where $a \neq 0$, $b$, and $c$ are real constants, and $g(x)$ is a forcing function. If $g(x) = 0$, the ODE is homogeneous; if $g(x) \neq 0$, it is non-homogeneous.

1. Solution Structure

The general solution $y(x)$ to a non-homogeneous equation consists of two distinct components:

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

• $y_h(x)$ (Homogeneous Solution): Solves $ay'' + by' + cy = 0$.
• $y_p(x)$ (Particular Solution): A specific solution satisfying the forcing function $g(x)$.

2. Homogeneous Case: The Characteristic Equation

Assuming a exponential trial solution $y = e^{rx}$, we obtain the quadratic characteristic equation:

$$ar^2 + br + c = 0$$

The nature of the roots $r_1, r_2$ is governed by the discriminant $\Delta = b^2 - 4ac$:

Discriminant	Root Nature	Homogeneous Solution $y_h(x)$
$\Delta > 0$	Real & Distinct ($r_1 \neq r_2$)	$$y_h(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}$$
$\Delta = 0$	Real & Repeated ($r_1 = r_2 = r$)	$$y_h(x) = (C_1 + C_2 x) e^{r x}$$
$\Delta < 0$	Complex Conjugates ($\alpha \pm i\beta$)	$$y_h(x) = e^{\alpha x} \left( C_1 \cos(\beta x) + C_2 \sin(\beta x) \right)$$

3. Non-Homogeneous Case: Undetermined Coefficients

When $g(x)$ consists of polynomials, exponentials, or sines/cosines, the Method of Undetermined Coefficients posits a trial form for $y_p(x)$ matching the terms in $g(x)$ and their derivatives.

Frequently Asked Questions

What happens during mechanical or electrical resonance?

Resonance occurs when the forcing frequency of $g(x)$ matches the natural frequency of the homogeneous system. In this case, the trial form for $y_p(x)$ must be multiplied by $x$ (or $x^2$) to preserve linear independence from $y_h(x)$.

How many initial conditions are required for a second-order ODE?

A second-order ODE requires two independent initial conditions (typically $y(x_0) = y_0$ and $y'(x_0) = y'_0$) to uniquely specify the arbitrary integration constants $C_1$ and $C_2$.

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