Mathematics · Differential Equations
Half-Life Differential Equation Calculator
Model exponential decay from an initial amount, half-life and elapsed time.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Half-lives elapsed = 18÷6=3.
- Decay factor = 0.5^3=0.125.
- Remaining amount = 80×0.125=10.
Problem → model → reason → result
What problem does this model solve?
Model exponential decay from an initial amount, half-life and elapsed time.
Why does the model apply?
Every complete half-life multiplies the remaining amount by one half; fractional half-lives follow the same exponential law.
What assumptions does it make?
The displayed differential equation represents the system, coefficients remain constant over the chosen interval, and the supplied initial condition selects one solution.
Formula
N(t)=N₀(1/2)ᵗ⁄ᴴ
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
An initial 80 units with half-life 6 after 18 time units leaves 10 units.
Common mistake
Half-life is not the time required for the whole amount to disappear.
When does this model not apply?
These closed-form and one-step numerical models do not capture changing coefficients, discontinuities, measurement uncertainty or every nonlinear system.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Half-life decay do?
Model exponential decay from an initial amount, half-life and elapsed time.
How does the Half-life decay work?
The calculator applies N(t)=N₀(1/2)ᵗ⁄ᴴ. Every complete half-life multiplies the remaining amount by one half; fractional half-lives follow the same exponential law.
What can I learn from the Half-life decay?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.