Calculation Result
Detailed Steps
Combined Solution Output
Calculator Inputs
Solution Graph
Values and Numerical Comparison
| Point | Symbolic solution | Euler estimate | Runge–Kutta estimate | Euler error | Runge–Kutta error |
|---|---|---|---|---|---|
| Calculate a solution to create the table. | |||||
Formula Used
A first-order linear equation has the standard form \(y' + P(x)y = Q(x)\). The integrating factor is \(\mu(x)=e^{\int P(x)\,dx}\). Multiplying creates an exact product derivative.
The general solution is \(y=\frac{\int \mu(x)Q(x)\,dx+C}{\mu(x)}\). An initial value determines the constant uniquely. Domain restrictions must remain valid throughout.
How to Use
- Select a complete equation or coefficient input mode.
- Enter the independent and dependent variable names.
- Add an initial condition when a particular solution is needed.
- Choose graph, precision, and numerical comparison options.
- Select Calculate Integrating Factor to display every step.
- Review verification, domain notes, graph curves, and errors.
- Copy, print, or download the complete calculation results.
Example Data
| Equation | Standard P | Standard Q | Suggested initial condition |
|---|---|---|---|
| y' + 2*y = 6 | 2 | 6 | (0, 1) |
| y' - (2/x)*y = x^2 | -2/x | x^2 | (1, 2) |
| x*y' + y = x^2 | 1/x | x | (1, 1) |
| y' + tan(x)*y = sin(x) | tan(x) | sin(x) | (0, 0) |
| y' + 3*y = exp(-x) | 3 | exp(-x) | (0, 2) |
| t*y' - 2*y = t^3 | -2/t | t^2 | (1, 0) |
Common Mistakes
Normalize the equation before identifying P and Q. Never skip division by the derivative coefficient. Check every excluded point before using initial data.
Use multiplication signs in complicated typed expressions. Keep symbolic variables separate from function names. Confirm that the equation remains linear in y.
Frequently Asked Questions
What equations can this calculator solve?
It solves first-order equations linear in the dependent variable. Coefficients may depend on the independent variable. Nonlinear powers of y require another method.
Why must the equation be normalized?
Normalization makes the derivative coefficient equal one. This reveals the correct P and Q functions. The integrating factor depends directly on normalized P.
What does the integrating factor accomplish?
It converts the left side into one product derivative. That derivative can be integrated in one step. This transformation produces the general solution efficiently.
Can I enter dy/dx instead of y'?
Yes, common derivative notations are accepted. The parser replaces them with an internal derivative symbol. Clear multiplication signs improve difficult equation parsing.
How is an initial condition used?
The initial point is substituted into the general solution. This calculates the integration constant from known data. The resulting expression is the particular solution.
Why can two answers look different?
Integration constants can absorb nonzero constant factors. Algebraic simplification can also change the visible expression. Equivalent answers satisfy the same differential equation.
What happens when symbolic integration fails?
The calculator keeps the integral in symbolic form. Numerical Euler and Runge–Kutta estimates may still work. Review the warning before trusting unsupported expressions.
Why are some graph points missing?
Undefined denominators or logarithms can create singular points. Large values may also exceed safe plotting limits. Missing points prevent misleading lines across discontinuities.
Does verification prove the solution is correct?
A zero symbolic residual strongly supports the result. Numerical residuals offer another useful consistency check. Always consider domains and original equation assumptions.