Differential Equation Solver
Solve ODEs step by step: first-order (separable, linear, homogeneous), second-order linear constant-coefficient, and initial-value problems — with exact symbolic solutions, a numerical (RK4) solver, slope fields and phase plots. Every symbolic answer is checked by back-substitution.
How to Use
- Type the equation using <code>y'</code> (or <code>dy/dx</code>) for the first derivative and <code>y''</code> for the second — e.g. <code>dy/dx = 3x^2</code> or <code>y'' + 3y' + 2y = 0</code>.
- For an <strong>initial-value problem</strong>, fill in x₀ and y(x₀) (and y′(x₀) for second-order) — or just write the conditions on their own lines, like <code>y(0) = 2</code>.
- The solver picks the method automatically: direct integration, separable, first-order linear (integrating factor), homogeneous (v = y/x), or the characteristic equation for second-order constant-coefficient equations.
- Every symbolic solution is <strong>verified by substituting it back into the equation</strong>; if it doesn't check out, the tool falls back to a numerical (RK4) solution instead of showing a wrong answer.
- The graph shows a <strong>slope field</strong> with your solution curve (first-order) and a <strong>phase plot</strong> of y vs y′ (second-order IVPs). Open <strong>Steps</strong> for the full working.
Slope field & solution
Methods it uses
About the Differential Equation Solver
Whether you are at a desk or on your phone, the Differential Equation Solver makes everyday maths and number work easy — and it is completely free. Solve ODEs step by step: first-order (separable, linear, homogeneous), second-order linear constant-coefficient, and initial-value problems — with exact symbolic solutions, a numerical (RK4) solver, slope fields and phase plots. Every symbolic answer is checked by back-substitution.
How it works
Type your numbers into the boxes. The answer shows up right away — you do not have to press a button. If you change a number, the answer changes too. So you can try different numbers and watch what happens, or check an answer you worked out yourself. Just make sure each box has the right kind of number in it.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
Which equations can it solve symbolically?
<strong>First-order:</strong> direct integration (y′ = f(x)), separable (y′ = g(x)·h(y)), first-order linear (y′ + P(x)y = Q(x), via the integrating factor μ = e^∫P), and homogeneous equations (y′ = F(y/x), via v = y/x). <strong>Second-order:</strong> linear with constant coefficients, a y″ + b y′ + c y = g(x) — the characteristic equation gives the homogeneous solution (distinct real, repeated, or complex roots) and undetermined coefficients give a particular solution for polynomial, exponential and sinusoidal forcing.
How do I know the answer is right?
Symbolic solutions are <strong>self-checked</strong>: the tool differentiates the solution it found and substitutes it back into the original equation at several points. Only solutions whose residual is essentially zero (and that match the initial conditions) are shown as “verified ✓”. If a form can't be verified, the tool reports a numerical solution rather than a possibly-wrong formula.
What's an initial-value problem (IVP)?
A differential equation plus enough conditions to pin down the arbitrary constants. A first-order equation needs one condition, y(x₀) = y₀; a second-order equation needs two, y(x₀) = y₀ and y′(x₀) = y₁. The solver applies them to determine C (or C₁ and C₂) and returns the single specific solution — for <code>dy/dx = 3x², y(0) = 2</code> that's <code>y = x³ + 2</code>.
What is the numerical solver?
A classical fourth-order Runge–Kutta (RK4) integrator. It works for <em>any</em> first-order equation with an initial condition — including ones with no closed-form solution, like the logistic equation — and for second-order equations by converting them to a first-order system. It also drives the slope-field solution curve and the phase plot.
What are slope fields and phase plots?
A <strong>slope field</strong> (direction field) draws a little line at each point (x, y) with slope f(x, y), so you can see the family of solution curves at a glance; the solver overlays your particular solution. A <strong>phase plot</strong> graphs y′ against y for a second-order system, revealing behaviour like oscillation (closed loops), damping (inward spirals) or growth.
Does it run online or send my equation away?
Everything runs locally in your browser — parsing, symbolic differentiation and integration, the solving methods, RK4 and the plots. Nothing is uploaded, and it works offline.
How do I use the Differential Equation Solver?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
ODE coursework
Check separable, linear and second-order solutions with the steps and a verification you can trust.
Physics & engineering
Damped/forced oscillators, RC/RL circuits and growth–decay models — symbolic or numeric.
Initial-value problems
Apply conditions to get the one specific solution, not just the general family.
Visualising behaviour
Slope fields and phase plots make stability, oscillation and damping obvious.
Models without closed forms
Logistic growth and nonlinear equations solved numerically with RK4.
Teaching
Show the method, the characteristic roots, and the solution curve together.
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