Solver Inputs
This guided solver returns analytical forms for standard ordinary differential equation families. Choose a model, enter coefficients, set initial conditions, and generate a table plus graph.
Example Data Table
| Model | Sample Equation | Initial Data | Representative Exact Solution |
|---|---|---|---|
| Exponential | y′ = 0.4y | y(0) = 2 | y = 2e^(0.4x) |
| Linear Constant | y′ + 2y = 6 | y(0) = 1 | y = 3 - 2e^(-2x) |
| Logistic | y′ = 0.8y(1 - y/50) | y(0) = 5 | y = 50 / (1 + 9e^(-0.8x)) |
| Second Order | y′′ + 2y′ + y = 0 | y(0) = 3, y′(0) = -1 | y = (3 + 2x)e^(-x) |
Formula Used
This calculator applies exact closed-form templates for standard ordinary differential equations. Each model uses a known analytical pathway:
- Exponential:
y′ = kygivesy = Ce^(kx). - Linear first order:
y′ + ay = bory′ + ay = be^(mx)uses an integrating factor. - Logistic:
y′ = ry(1-y/K)is solved by separation and partial fractions. - Newton cooling:
T′ = -k(T-Ts)is reduced to exponential relaxation. - Bernoulli:
y′ + ay = by^nbecomes linear afterv = y^(1-n). - Second order:
y′′ + py′ + qy = 0is solved from the characteristic equationr² + pr + q = 0.
How to Use This Calculator
- Select the ODE family matching your differential equation.
- Enter coefficients such as
a,b,k,m,K,p, orq. - Provide initial conditions at
x₀. Second-order mode also needsy′(x₀). - Set the plotting interval and number of points.
- Press Solve Analytical ODE to display the exact solution above the form.
- Review the generated table and graph, then export the data as CSV or PDF.
Frequently Asked Questions
1) Does this solve any differential equation?
No. It solves several important analytical families with exact formulas. That keeps the output reliable, readable, and suitable for study, checking homework, or quick engineering-style calculations.
2) Why are some inputs hidden?
Each ODE family needs different coefficients. The page only shows the fields relevant to the selected method so the form stays clean and easier to use.
3) What does the initial point x₀ do?
It anchors the integration constant. Instead of returning a generic constant C only, the calculator uses y(x₀) and sometimes y′(x₀) to produce one concrete exact solution.
4) What happens in second-order mode?
The solver forms the characteristic equation, checks the discriminant, and then uses the matching real-root, repeated-root, or complex-root formula with your initial conditions.
5) Why can Bernoulli output become undefined?
Some Bernoulli parameter sets produce expressions that leave the real-number domain for parts of the chosen x-range. In such cases, the plot and table show missing or undefined values.
6) Is the graph based on the exact formula?
Yes. The chart samples points directly from the analytical solution returned by the selected model. It is not a separate numerical approximation routine.
7) What should I use for growth and saturation models?
Use exponential mode for unrestricted growth or decay. Use logistic mode when growth slows near a carrying capacity or practical upper limit.
8) Can I export my results?
Yes. After solving, use the CSV button for spreadsheet-friendly data and the PDF button for a printable summary of the generated results table.