2nd Order Nonlinear Differential Equation Solver Calculator

Calculator Input

Equation type

Start value x₀

End value x_end

Step size h

Initial y(x₀)

Initial y′(x₀)

δ Damping

Linear damping on y′

α Linear stiffness

Coefficient on y

β Cubic stiffness

Coefficient on y³

γ Forcing amplitude

External cosine force

ω Driving frequency

Frequency of forcing

b Constant bias

Constant acceleration shift

η Sine nonlinearity

Extra sin(y) term

y'' = 0.3·cos(1.2x) + 0 - 0.2y' - -1y - 1y³ + 0sin(y)

Example Data Table

Model	x_end	h	y₀	y′₀	Key coefficients
Duffing Oscillator	30	0.05	1.0	0.0	δ=0.20, α=-1, β=1, γ=0.30, ω=1.20
Van der Pol Oscillator	25	0.04	0.5	0.0	μ=1.20, ω₀=1, A=0.20, k=1
Damped Nonlinear Pendulum	20	0.03	0.8	0.0	c=0.15, g=9.81, L=1, F=0.35, ω=1.30
Custom Nonlinear Form	12	0.02	0.3	0.1	a=0.20, b=-0.10, c=0.05, d=0.40, k=1.50, e=0.30

Formula Used

General second-order nonlinear equation

y″ = f(x, y, y′)

State conversion

Let y₁ = y and y₂ = y′.

Then y₁′ = y₂ and y₂′ = f(x, y₁, y₂).

Fourth-order Runge–Kutta update

y_n+1 = y_n + (k₁ + 2k₂ + 2k₃ + k₄) / 6

v_n+1 = v_n + (m₁ + 2m₂ + 2m₃ + m₄) / 6

Pseudo-energy indicator

E* = 0.5(y′)² + 0.5y²

The calculator transforms the second-order equation into two first-order equations. It then applies the fourth-order Runge–Kutta method across the chosen x-range. This gives stable, high-quality approximations for many nonlinear systems.

The pseudo-energy is a general indicator only. It helps compare relative motion intensity. It is not the exact physical energy for every model.

How to Use This Calculator

Select the nonlinear model that matches your problem.
Enter the start value, end value, and step size.
Fill in the initial displacement y(x₀).
Fill in the initial velocity y′(x₀).
Enter the model coefficients shown beside each parameter field.
Press Solve Equation to compute the solution.
Review the result summary, graph, phase plot, and preview table.
Use the CSV or PDF buttons to export the output.

FAQs

1) What does this calculator solve?

It solves several second-order nonlinear differential equations numerically. Included options are Duffing, Van der Pol, damped pendulum, and a custom nonlinear form. It is useful when a simple closed-form solution is unavailable.

2) Why is a numerical method used here?

Many nonlinear equations cannot be solved exactly with elementary formulas. A numerical method steps through the interval and builds an accurate approximation. This makes the tool practical for realistic oscillation and forcing problems.

3) What method powers the solver?

The page uses the classical fourth-order Runge–Kutta method. It converts the second-order equation into two first-order equations first. Then it updates displacement and velocity together at each step.

4) How should I choose the step size?

Smaller step sizes usually improve accuracy but increase runtime. Start with a moderate value like 0.02, 0.03, or 0.05. If the graph looks rough or unstable, reduce the step size.

5) What do y, y′, and y″ mean?

y is the state or displacement. y′ is the first derivative, often interpreted as velocity. y″ is the second derivative, often interpreted as acceleration or system curvature.

6) Is the pseudo-energy the exact physical energy?

No. It is a general comparison measure used to track motion intensity. It is helpful for trend inspection, but model-specific physical energy may require a different expression.

7) Can I solve any arbitrary symbolic nonlinear equation?

This page does not include a full symbolic parser. It provides advanced preset models and a flexible custom coefficient form. That keeps the interface fast, secure, and easy to validate.

8) What should I do if the result looks unstable?

Reduce the step size first. Then check whether the coefficients are too large for the selected interval. Highly nonlinear systems can diverge quickly when forcing or stiffness values are extreme.