Solve and visualize first and second-order ODEs numerically. Supports Euler, RK4 methods with phase portraits.
Explore an initial-value problem by entering an ordinary differential equation, its starting conditions, an interval, and a numerical step size. The solver advances the solution with either Euler's method or fourth-order Runge-Kutta (RK4), then visualizes the resulting trajectory and, where applicable, its phase portrait. The main surprise is that the result is not a symbolic formula. It is a sequence of approximated values whose accuracy depends strongly on the chosen method and step size.
A concrete input and expected output from the current implementation.
Input
Equation: y' = y Initial condition: y(0) = 1 Interval: 0 to 0.2 Step size: 0.1 Method: Euler
Expected output
x = 0, y = 1 x = 0.1, y = 1.1 x = 0.2, y = 1.21
Euler's update is y_next = y + h f(x,y). The first step gives 1 + 0.1(1) = 1.1, and the second gives 1.1 + 0.1(1.1) = 1.21.
Euler's method evaluates the derivative at the current solution point and follows that slope for one full step. For y' = f(x,y), each update is y_next = y + h f(x,y). It is easy to inspect, but its accumulated error can be conspicuous when the interval is long, the step is large, or the solution changes rapidly.
RK4 samples four slopes during each step and combines them into one update, usually giving a closer approximation than Euler at the same step size. A second-order equation can be viewed as two linked first-order equations by setting v = y'. This state representation supports a phase portrait showing y against its derivative rather than against the independent variable.
No. Euler and RK4 generate numerical approximations from an initial condition. They can show how a solution develops over the requested interval, but they do not prove an identity or return a general formula containing arbitrary constants. Use a symbolic differential-equation system when an exact form is required.
They approximate each step differently. Euler uses one slope from the start of the step, while RK4 combines four slope evaluations. Their results should generally move closer as the step size is reduced for a well-behaved problem. Persistent divergence can indicate an inadequate step size, instability, a singularity, or an incorrectly entered equation or initial condition.
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