Mathematics · Differential Equations
Exponential Growth and Decay ODE Calculator
Solve y′=ky from an initial value and evaluate the solution at time t.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Multiply rate and time: 0.05×10=0.5.
- Compute e^0.5=1.6487212707001282.
- y(10)=100×1.6487212707001282=164.87212707001282.
Problem → model → reason → result
What problem does this model solve?
Solve y′=ky from an initial value and evaluate the solution at time t.
Why does the model apply?
When the rate of change is proportional to the current amount, separating variables produces an exponential solution.
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
y(t)=y₀eᵏᵗ
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
For y₀=100, k=0.05 and t=10, y≈164.87.
Common mistake
k is a continuous rate; a 5% discrete period multiplier is a different model.
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 Exponential growth ODE do?
Solve y′=ky from an initial value and evaluate the solution at time t.
How does the Exponential growth ODE work?
The calculator applies y(t)=y₀eᵏᵗ. When the rate of change is proportional to the current amount, separating variables produces an exponential solution.
What can I learn from the Exponential growth ODE?
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.