Enter an equation — the calculator detects its type, shows the solution step by step with the integrals and finds the general solution and the IVP solution. First-order (separable, linear, Bernoulli, homogeneous), second-order linear with constant coefficients and numerical methods.
Linear equations with constant coefficients
a·y'' + b·y' + c·y = f(x) is solved as y = y_h + y_p: the homogeneous solution plus any particular solution.
Homogeneous equation
Substituting y = e^(rx) gives a·r² + b·r + c = 0. Distinct real roots — C₁e^(r₁x) + C₂e^(r₂x); repeated — (C₁ + C₂x)e^(rx); complex α ± iβ — e^(αx)(C₁cos βx + C₂sin βx).
Undetermined coefficients
For f(x) = e^(αx)(Pₙ(x)cos βx + Qₘ(x)sin βx) try the same form with unknown polynomial coefficients; if α + iβ is a root of multiplicity s, multiply by xˢ.
Numerical methods
Euler is first-order accurate, improved Euler second-order, RK4 fourth-order. Halve the step to check convergence.
FAQ
How is a second-order linear non-homogeneous ODE solved?
Find the homogeneous solution from the characteristic roots, guess a particular solution from the right-hand side and add them.
What is resonance?
When α + iβ of the right side is a characteristic root; the trial form is multiplied by xˢ.
Which right-hand sides are supported?
Polynomials, exponentials, sines and cosines, their products and sums. Others need variation of parameters — use the numerical tab.
Why is Runge–Kutta better than Euler?
Euler error is O(h), RK4 error is O(h⁴).
Which first-order equations are solved analytically?
Separable, linear y' + p(x)y = q(x), Bernoulli and homogeneous y' = φ(y/x). Antiderivatives are verified by differentiation.