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Differential Equations

First and second order ODEs, initial value problems, Laplace transforms — exact symbolic solutions with every step shown.

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What Zeretis can solve

Zeretis solves ordinary differential equations symbolically — returning the exact general or particular solution, not a numerical approximation. It identifies the equation type automatically and applies the appropriate method.

First order ODEs

Separable, linear, exact, Bernoulli, homogeneous — method identified automatically.

Second order ODEs

Constant coefficient, variation of parameters, undetermined coefficients.

Initial value problems

Provide initial conditions and get the particular solution with constants resolved.

Laplace transforms

Forward and inverse transforms, solving ODEs via the Laplace method.

Bernoulli equations

Non-linear first order ODEs reducible via substitution.

Systems of ODEs

Coupled first-order systems, solved via eigenvalue method.

How to type ODE problems

Separable ODE

solve dy/dx = x * y

→ y = Ce^(x²/2)

Initial value problem

solve dy/dx = 2x, y(0) = 3

→ y = x² + 3

Second order

solve y'' - 3y' + 2y = 0

→ y = C₁eˣ + C₂e²ˣ

Laplace transform

Laplace transform of t^2 * e^(3t)

→ 2 / (s−3)³

Non-homogeneous second order

solve y'' + y = sin(x), y(0) = 0, y'(0) = 1

→ y = (1/2)(sin x − x cos x) + sin x

Best practices

Use y', y'', or dy/dx, d²y/dx² — all are understood
For initial conditions, append them naturally: "y(0) = 1, y'(0) = 0"
The solver identifies the ODE type automatically — you don't need to specify "separable" or "linear"
For Laplace transforms of piecewise functions, describe the pieces clearly
Constants of integration are labelled C₁, C₂ etc. — particular solutions have these resolved
For systems: "solve the system: x' = x + y, y' = x - y"

📐Algebra ∫Calculus 🔢Linear Algebra ∑Series & Sequences 📏Trigonometry

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