Differential Equation Solver
Enter a first-order or second-order ordinary differential equation and receive the algebraic method, intermediate steps, general solution, condition-specific solution and numerical verification when supported. A separate workspace handles higher derivatives, partial derivatives and common PDE classification.
Enter the differential equation
Accepted examples include y' = -2*y + 4, dy/dx = x*y, y'' + 4*y = 0 and implicit derivative equations.
Verified equation examples
Step-by-step differential equation solution
Enter an equation and select Solve step by step.
Verification graph
Secondary visualization of the solved result
Waiting for an equation
| x | y | y′ | Estimated local comparison |
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Ordinary and partial derivative calculator
Enter an expression rather than a complete differential equation. Select the differentiation variable and order.
Derivative result
Partial differential equation analyzer
Enter common notation such as u_t = alpha*u_xx, u_tt = c^2*u_xx or u_xx + u_yy = 0.
PDE classification
What the solver can determine
The engine identifies derivative order, whether the equation is ordinary or partial, whether a first- or second-order equation can be evaluated as an initial-value problem, and whether it belongs to a built-in exact family. It recognizes constant-coefficient first-order linear equations, exponential growth or decay, logistic equations, direct-integration forms and homogeneous second-order constant-coefficient equations.
When numerical solving is required
Many nonlinear or variable-coefficient differential equations do not reduce to an elementary formula. For supported first- and second-order forms, the calculator isolates the derivative numerically at each step and integrates the initial-value problem using Euler, Heun or fourth-order Runge–Kutta. A step-halving comparison estimates numerical sensitivity.
Initial and boundary information
A first-order initial-value problem needs one condition such as y(0)=1. A second-order problem normally needs two conditions, such as y(0)=1 and y′(0)=0. General boundary-value problems can require a shooting or finite-difference method and are not silently treated as initial-value problems.
Exact, implicit and approximate answers
A mathematically honest solver must distinguish an elementary exact solution, an implicit relation, a special-function or series solution, and a numerical approximation. This calculator reports the available class and does not manufacture an exact form when the equation is outside its verified analytic library.
Accepted differential notation
First derivative
y', dy/dx or the internal symbol yp.
Second derivative
y'', d2y/dx2, d^2y/dx^2 or ypp.
Functions
sin, cos, tan, exp, ln, sqrt, abs and common inverse or hyperbolic functions.
PDE tokens
u_t, u_tt, u_x, u_xx, u_y, u_yy for classification of common forms.
Differential equation solver questions
Can this solve every differential equation exactly?
No mathematical system can express every differential equation in elementary closed form. This calculator gives exact-family solutions where verified and numerical IVP solutions where the derivative can be evaluated reliably.
Does it support second derivatives?
Yes. Enter y double prime notation and provide both y(x₀) and y′(x₀). The solver treats the equation as a first-order system for numerical integration.
Can I calculate a partial derivative without solving a PDE?
Yes. Use the derivative workspace, enter an expression containing several variables, and select the variable and derivative order.
Are equations sent to an external server?
No. Parsing, symbolic differentiation, numerical integration and plotting data preparation run in the browser. The packaged plotting library is loaded locally.