Differential Equation Calculator
Find general solutions for common ODE forms used in calculus and engineering: second-order ay'' + by' + cy = 0, first-order y' + py = q, and y' = ky.
FAQ for this calculator
- What ODE types are supported?
- Second-order homogeneous with constant coefficients, first-order linear y' + py = q (constant p, q), and y' = ky.
- Can I enter initial conditions?
- Not yet—this tool gives the general solution with C₁, C₂; substitute x₀ and y₀ to solve for constants.
- What about non-homogeneous second order?
- Not in this version—only ay'' + by' + cy = 0 for second order.
- Are complex roots handled?
- Yes—complex conjugate roots α ± βi produce e^(αx)(C₁ cos βx + C₂ sin βx).
How to use the differential equation calculator
Second-order constant-coefficient ODEs use the characteristic polynomial ar² + br + c = 0; real distinct, repeated, or complex roots yield different solution forms.
- Choose second-order, first-order linear, or exponential.
- Enter coefficients from your homework problem.
- Read the general solution and apply initial conditions separately if needed.
When to use this calculator
- Checking characteristic roots for ay'' − 3y' + 2y = 0.
- First-order linear y' + 2y = 6 equilibrium and transient.
- Radioactive decay y' = −λy exponential form.
Examples & walkthrough
- y'' − 3y' + 2y = 0 → roots 1 and 2 → y = C₁eˣ + C₂e²ˣ.
- y' + 2y = 6 → y = 3 + Ce⁻²ˣ.