Initial Value Problem Calculator
Solve first-order IVPs easily: enter differential equation, initial condition, and target point to compute accurate solutions with clear steps quickly.
Equation Preview
Helping Notes
- We solve the IVP numerically with an adaptive Runge–Kutta–Fehlberg (RK45) method and tight error control.
- Typical IVP tools require f(x,y), the initial condition (x₀,y₀), and an x to evaluate y(x). :contentReference[oaicite:1]{index=1}
- Enter expressions using math.js syntax (sin(x), exp(x), x^2, etc.).
- If f or y becomes undefined, we’ll show an error at the step where it happened.
Results
Solution at Target
Computation Summary
Sample Points
Error
What is Initial Value Problem Calculator?
An Initial Value Problem (IVP) asks for a function satisfying a differential equation together with specified starting values. Given a differential equation and a point \((t_0, y_0)\), the goal is to find \(y(t)\) such that the equation holds and \(y(t_0)=y_0\). Many physical, biological, and financial models start this way—motion with initial position/velocity, population with initial size, temperature with initial state. A general IVP is written:
When conditions ensure existence and uniqueness (e.g., continuity in \(t\) and Lipschitz in \(y\)), the solution is well-defined near \(t_0\). Linear first‑order IVPs take the form:
About the Initial Value Problem Calculator
This calculator solves IVPs symbolically when possible and numerically when necessary. It supports separable, linear, and many standard nonlinear models, and can treat systems by converting higher‑order equations to first‑order vectors. It shows each step: classification, algebraic manipulation (e.g., integrating factor or separation), and a clear record of numerical updates when exact forms are unavailable. For numerical work, common schemes include explicit Euler, improved Euler (Heun), and classical Runge–Kutta of order four (RK4):
It also reports local/global error orders and warns about stiffness or discontinuities.
How to Use this Initial Value Problem Calculator
- Enter the differential equation (e.g., y' = y − t) and initial condition (e.g., y(0)=1).
- Select solution mode (symbolic, Euler, Heun, or RK4) and step size \(h\) for numeric methods.
- Click calculate to compute \(y(t)\) or a table of \((t_n,y_n)\) values with intermediate steps.
- Review responsive formulas and, if desired, request linearization or error estimates.
- Copy the result and steps for homework, labs, or engineering verification.
Examples
Example 1: Separable IVP
\(y' = y,\; y(0)=1\)
Example 2: Linear with integrating factor
\(y'+2y=0,\; y(0)=3\)
Example 3: Numerical RK4 step
\(y'=t,\; y(0)=2\) with step \(h=0.5\) from \(t_0=0\) to \(t_1=0.5\).
Example 4: Logistic growth
\(y'=r y(1-\tfrac{y}{K}),\; y(0)=y_0\)
FAQs
What is an initial value problem?
An ODE with conditions at one point, seeking a solution that satisfies both the equation and the initial data.
Which numerical methods are available?
Explicit Euler, Heun (improved Euler), and RK4; each shows its update formulas and error order.
Does it handle systems or higher‑order equations?
Yes. Higher‑order ODEs are rewritten as first‑order systems; you can input vector equations.
How do I choose a step size?
Start modest (e.g., \(h\) small) and refine until results stabilize; RK4 usually allows larger \(h\) for similar accuracy.
What is the difference between local and global error?
Local error is per step; global error accumulates over many steps. RK4 is \(O(h^4)\) globally under smoothness.
Can it detect stiffness?
It warns when explicit methods behave poorly; stiff problems often need implicit solvers (outside scope here).
Will it show exact solutions when possible?
Yes. For separable and linear cases, it derives closed forms and verifies the initial condition.
Can I use parameters?
Yes. Enter symbols like \(r\) and \(K\); results display symbolically alongside numeric evaluations if values are provided.
What input formats are supported?
Derivative as y' or dy/dt, initial data as y(t0)=y0; systems as comma‑separated component equations.
How do I verify correctness?
Differentiate the solution and check the IVP, or compare numeric tables across smaller step sizes.