Second-Order ODE Solver
Solve and visualize homogeneous constant coefficient equations of the form ay” + by’ + cy = 0.
Coefficients
Initial Conditions
Characteristic Eq
ar² + br + c = 0
Roots ($r_1, r_2$) determine the solution type: Real distinct, Real repeated, or Complex conjugates.
Solution Details
Calculating…Solution Curve y(t)
Understanding Second-Order Homogeneous ODEs
Second-order linear differential equations with constant coefficients appear everywhere in physics and engineering, most notably in Mechanical Vibrations (mass-spring-damper systems) and Electrical Circuits (RLC circuits). The behavior of the system is governed entirely by the roots of the Characteristic Equation: $$ ar^2 + br + c = 0 $$
Two distinct real roots $r_1, r_2$. The system returns to equilibrium without oscillating. Think of a door closing slowly with a heavy hydraulic damper.
One repeated real root $r$. This is the fastest way to return to equilibrium without overshooting. Essential for car suspension design.
Complex conjugate roots $\alpha \pm i\beta$. The system oscillates with decaying amplitude. Think of a guitar string vibrating or a pendulum swinging.
In this tool, we solve for the unknown constants $c_1$ and $c_2$ using the user-provided initial conditions $y(0)$ (initial position) and $y'(0)$ (initial velocity). This creates a unique “Particular Solution” which is plotted above.
Frequently Asked Questions
What if ‘a’ is zero?
If $a=0$, the equation becomes $by’ + cy = 0$, which is a First-Order Linear ODE. This tool is designed specifically for second-order equations ($a \neq 0$). If you enter $a=0$, the solver may behave unpredictably or show an error, as division by zero occurs in the quadratic formula.
What is the physical meaning of negative ‘b’?
In a physical system, $b$ represents damping (friction/resistance). A positive $b$ removes energy from the system. A negative $b$ would imply “negative damping,” meaning energy is being added to the system continuously. This leads to unstable solutions where the amplitude grows to infinity over time.
Why does the graph stop?
The graph plots $y(t)$ for a finite time range (typically $t=0$ to $t=20$ or similar). While the mathematical solution continues forever, we only display the relevant transient behavior where the most interesting dynamics occur.