Ordinary Differential Equation Solver Calculator
Work out ordinary differential equation solver instantly with clear inputs, formula shown and shareable results.
About Ordinary Differential Equation Solver Calculator
The linear equation dy/dx = ay + b has the exact solution y = (y₀ + b/a)e^(ax) − b/a. When a is negative the exponential decays and y approaches the steady state −b/a.
Formula
Linear ODE solution
y(x) = (y₀ + b/a)·e^(ax) − b/a, for a ≠ 0
Worked example
With the default values below, Ordinary Differential Equation Solver Calculator returns y at your x (exact) of 3.59399415. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficient a in dy/dx = a·y + b
- -0.5
- Constant b
- 2
- Initial value y(0)
- 1
- Solve at x =
- 4
Results
- y at your x (exact)
- 3.59399415
- Euler's method estimate
- 3.59440022
- Steady state −b/a
- 4.00000000
- dy/dx there
- 0.20300292
Step by step
- 1Values used
- 2Linear ODE solution
- 3y at your x (exact)= 3.59399415
- 4Euler's method estimate= 3.59440022
- 5Steady state −b/a= 4.00000000
- 6dy/dx there= 0.20300292
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For dy/dx = −0.5y + 2 with y(0) = 1 at x = 4?
The steady state is 4, and y(4) = (1 − 4)e⁻² + 4 ≈ 3.5939942.
What does the steady state mean?
It is the value where dy/dx = 0, so the system stops changing — stable when a is negative.