Second-Order Homogeneous Differential Equation Calculator
Our online calculator provides a complete, step-by-step solution for homogeneous second-order linear differential equations with constant coefficients. Master the characteristic equation and find the general solution for all types of roots.
Homogeneous DE Solver
Input your equation to get a complete solution, including the characteristic equation, roots, and the general solution with detailed explanations.
Homogeneous DE Concepts
Master the fundamental concepts behind solving homogeneous differential equations, including the derivation of the characteristic equation and the theory of superposition.
Interactive DE Solver & Plotter
Solve $ay'' + by' + cy = 0$
Initial Conditions (Optional)
Understanding (Second-Order) 2nd Order Differential Equations
A second-order linear homogeneous differential equation with constant coefficients is a cornerstone in the study of differential equations. It takes the general form:
Alternatively, using prime notation ($y' = \frac{dy}{dx}$, $y'' = \frac{d^2y}{dx^2}$):
Key characteristics of this equation type:
- 2nd Order (Second Order): The highest derivative present is the second derivative ($y''$).
- Linear: The dependent variable $y$ and its derivatives ($y', y''$) appear only to the first power and are not multiplied together.
- Homogeneous: The right-hand side of the equation is zero. If it were a non-zero function of $x$, say $f(x)$, the equation would be non-homogeneous.
- Constant Coefficients: The coefficients $a, b,$ and $c$ are real numbers (constants), with the crucial condition that $a \neq 0$ (otherwise, it wouldn't be a second-order equation).
These equations are incredibly powerful for modeling a wide array of physical phenomena. For instance, they describe the motion of a mass attached to a spring (simple harmonic motion, damped oscillations), the flow of charge in an RLC electrical circuit, and many other systems that exhibit oscillatory or exponential decay/growth behavior.
The Auxiliary (Characteristic) Equation Method
The standard approach to solving $ay'' + by' + cy = 0$ involves a clever substitution. We propose a solution of the exponential form $y = e^{mx}$, where $m$ is a constant to be determined.
Step 1: Assume a Solution Form
Let $y = e^{mx}$. Then, the derivatives are:
- $y' = \frac{dy}{dx} = me^{mx}$
- $y'' = \frac{d^2y}{dx^2} = m^2e^{mx}$
Step 2: Substitute into the DE
Plugging these into the original differential equation $ay'' + by' + cy = 0$ gives:
We can factor out $e^{mx}$ (which is never zero for any real $m$ and $x$):
Step 3: Form the Auxiliary Equation
Since $e^{mx} \neq 0$, for the equation to hold, the term in the parenthesis must be zero. This gives us the auxiliary equation (also known as the characteristic equation):
This is a simple quadratic equation in $m$. The roots of this auxiliary equation dictate the form of the general solution to the differential equation.
Step 4: Find the Roots of the Auxiliary Equation
We use the quadratic formula to find the roots $m_1$ and $m_2$:
The nature of these roots depends on the discriminant, $\Delta = b^2 - 4ac$. There are three distinct cases:
Case 1: Real and Distinct Roots ($\Delta > 0$)
If $b^2 - 4ac > 0$, the auxiliary equation has two distinct real roots, $m_1$ and $m_2$. In this scenario, $e^{m_1x}$ and $e^{m_2x}$ are two linearly independent solutions. The general solution is a linear combination of these:
Where $A$ and $B$ are arbitrary constants. These constants are typically determined if initial conditions (e.g., $y(x_0) = y_0$ and $y'(x_0) = y'_0$) are provided.
Case 2: Real and Equal Roots ($\Delta = 0$)
If $b^2 - 4ac = 0$, the auxiliary equation has one real root (a repeated root), $m_1 = m_2 = m = -\frac{b}{2a}$. One solution is $y_1 = e^{mx}$. A second, linearly independent solution can be found (e.g., using the method of reduction of order) to be $y_2 = xe^{mx}$. The general solution is then:
$A$ and $B$ are arbitrary constants.
Case 3: Complex Conjugate Roots ($\Delta < 0$)
If $b^2 - 4ac < 0$, the auxiliary equation has two complex conjugate roots. These roots can be written as $m_1 = \alpha + i\beta$ and $m_2 = \alpha - i\beta$, where:
- $\alpha = -\frac{b}{2a}$ (the real part)
- $\beta = \frac{\sqrt{4ac - b^2}}{2a}$ (the imaginary part, $\beta > 0$)
The two linearly independent solutions would initially appear as $e^{(\alpha+i\beta)x}$ and $e^{(\alpha-i\beta)x}$. Using Euler's formula ($e^{i\theta} = \cos\theta + i\sin\theta$), these can be rewritten in terms of real-valued functions. The general solution becomes:
$A$ and $B$ are arbitrary constants. This form of the solution often represents damped oscillations in physical systems.
Step 5: Apply Initial Conditions (if given)
If initial conditions such as $y(x_0) = y_0$ and $y'(x_0) = y'_0$ are provided, they are used to solve for the constants $A$ and $B$ in the general solution, yielding a unique particular solution.