The three equations
dy/dt = k y gives exponential growth or decay, with y = y₀e^(kt).
dy/dt = −k(y − T) is Newton's cooling, heading towards the surrounding temperature. dy/dt = k y(1 − y/L) is logistic growth, flattening off at the capacity L.
Worked example
Start at 100 with k = 0.1 and ask for the value at t = 10. The solution is y = 100e^(0.1t), so y = 271.828183.
The doubling time is 6.931472, and a Runge–Kutta run with 1,000 steps gives the same 271.828183.
Why both answers are shown
The closed form is exact and fast. The numerical run makes no assumption about the shape, so when the two agree the solution is confirmed.
If they ever disagreed, the closed form would be wrong and you would see it immediately.
What it does not do
It solves these three forms, not an arbitrary equation you type in. Solving a general dy/dx = f(x, y) symbolically needs a computer algebra system.
Second-order equations, such as those for springs and circuits, are outside its range.
Inputs for the differential equation calculator
Which equation?
Pick growth and decay, Newton's cooling, or logistic growth.
Rate constant
How fast the change happens. A negative value turns growth into decay.
Starting value
The amount at time zero, written y(0).
Time to report
The moment you want the solution at. The curve is drawn to 1.5 times this.
CALCULATOR QUESTIONS
Differential equation solution calculator questions
What is the solution of dy/dt = 0.1y starting at 100?
y = 100e^(0.1t). At t = 10 that is 271.828183.
How do I find a doubling time?
It is ln(2) / k. For k = 0.1 that is 6.931472.
What is Newton's cooling?
The rate of cooling is proportional to how far the object is from the surrounding temperature, so it approaches that temperature without reaching it.
Can it solve a second-order equation?
No. This page handles first-order equations in these three standard forms.