Homogeneous differential equation & ODE solver

For the second-order homogeneous equation with constant coefficients, ay'' + by' + cy = 0, substitute the trial solution e^(rx) to obtain the characteristic polynomial a r² + b r + c = 0.

The roots decide the general solution: two distinct real roots → y = C₁ e^(r₁x) + C₂ e^(r₂x); a repeated root r → y = (C₁ + C₂ x) e^(rx); complex conjugates α ± βi → y = e^(αx)(C₁ cos βx + C₂ sin βx).

“General solution” means the family of solutions with arbitrary constants; initial conditions pin those constants down afterward.

a r² + b r + c = 0  →  y_h(x) from the root case

What is a homogeneous differential equation solver doing here?

For second order it solves ay'' + by' + cy = 0—the homogeneous constant-coefficient case—via the characteristic equation and returns the general solution.

What ODE types are supported?

Second-order homogeneous with constant coefficients, first-order linear y' + py = q (constant p, q), and y' = ky.

What does “general solution” mean?

The full family of solutions with arbitrary constants (C₁, C₂, …). Particular solutions need initial or boundary conditions applied afterward.

Can I enter initial conditions?

Not yet—the tool prints the general solution; substitute x₀ and y₀ (and y'₀ for second order) to solve for constants by hand or CAS.

What about non-homogeneous second order?

Not in this version—only ay'' + by' + cy = 0 for second order. Non-homogeneous problems need an extra particular solution.

Are complex roots handled?

Yes—complex conjugates α ± βi produce e^(αx)(C₁ cos βx + C₂ sin βx).

Choose the ODE type that matches your problem, enter coefficients, and read the general solution. Second-order mode is the homogeneous constant-coefficient solver most calculus courses start with.

• Select second-order homogeneous, first-order linear, or exponential growth/decay.
• Enter coefficients exactly as written in the textbook (signs matter).
• Copy the general solution, then apply y(x₀)=y₀ (and y' if needed) to solve for constants.
• For complex roots, keep the real trigonometric form shown by the tool.

• Checking characteristic roots for ay'' − 3y' + 2y = 0 (homogeneous).
• First-order linear y' + 2y = 6 equilibrium and transient.
• Radioactive decay y' = −λy exponential form.
• Confirming whether a homework ODE is homogeneous before trying undetermined coefficients.

1. Homogeneous: y'' − 3y' + 2y = 0 → roots 1 and 2 → y = C₁eˣ + C₂e²ˣ.
2. Repeated root: y'' − 4y' + 4y = 0 → r = 2 → y = (C₁ + C₂ x) e²ˣ.
3. Complex roots: y'' + 4y' + 13y = 0 → −2 ± 3i → y = e^(−2x)(C₁ cos 3x + C₂ sin 3x).
4. First-order linear: y' + 2y = 6 → y = 3 + Ce⁻²ˣ.

Same second-order homogeneous template; different characteristic roots change the general solution shape.