How to use this calculator

Enter the constants p and q and the starting value y(0). This form models RC circuits, Newton’s law of cooling, mixing tanks and terminal velocity with linear drag.

Method

Standard formy′ + py = q

Integrating factorμ = e^pt (μy)′ = qμ

Solutiony = qp + (y₀ − qp) e^−pt

Worked example

y′ + 0.5y = 10, y(0) = 2 (the default)

μ = e^0.5t, so (e^0.5ty)′ = 10e^0.5t.
Integrate: e^0.5ty = 20e^0.5t + C, so y = 20 + Ce^−0.5t.
y(0) = 2 gives C = −18: y = 20 − 18e^−0.5t.
y(3) = 20 − 18e^−1.5 = 15.98.

Frequently asked questions

What if p or q depends on t?

The integrating factor becomes e^{∫p dt} and the integral may need to be done by hand or numerically. Try the Euler’s method calculator for a numerical answer.

What is the time constant?

τ = 1/p. The solution covers 63% of the gap to its steady value q/p every τ.