First Order Linear ODE Calculator
Work out first order linear ode instantly with clear inputs, formula shown and shareable results.
About First Order Linear ODE Calculator
Multiplying dy/dx + py = q by the integrating factor e^(px) makes the left side an exact derivative. Integrating gives y = q/p + (y₀ − q/p)e^(−px), a steady part plus a decaying transient.
Formulas
Integrating factor
multiply by e^(∫p dx), then d/dx(y·e^(px)) = q·e^(px)
Solution
y = q/p + (y₀ − q/p)·e^(−px)
Worked example
With the default values below, First Order Linear ODE Calculator returns y at your x of 2.72932943. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Coefficient p in dy/dx + p·y = q
- 2
- Right-hand side q
- 6
- Initial value y(0)
- 1
- Evaluate at x =
- 1
Results
- y at your x
- 2.72932943
- Particular (steady) solution q/p
- 3.00000000
- Integrating factor e^(px)
- 7.38905610
- Time constant 1/p
- 0.50000000
Step by step
- 1Values used
- 2Integrating factor
- 3Solution
- 4y at your x= 2.72932943
- 5Particular (steady) solution q/p= 3.00000000
- 6Integrating factor e^(px)= 7.38905610
- 7Time constant 1/p= 0.50000000
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 + 2y = 6 with y(0) = 1 at x = 1?
The steady value is 3, so y = 3 − 2e⁻² ≈ 2.7293294.
What is the time constant for?
After 1/p the transient has shrunk to about 37% of its initial size, and after five time constants it is essentially gone.