Mathematics · Differential Equations
First-Order Linear ODE Calculator
Solve y′+py=q with constant coefficients and an initial value.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Equilibrium = 10÷2=5.
- Transient = (1−5)e^(−2×1)=-0.5413411329464508.
- y(1)=5+-0.5413411329464508=4.458658867053549.
Problem → model → reason → result
What problem does this model solve?
Solve y′+py=q with constant coefficients and an initial value.
Why does the model apply?
The solution combines the equilibrium q/p with a transient exponential term that carries the initial condition.
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)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0
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 y′+2y=10, y(0)=1, the solution at t=1 is about 4.46.
Common mistake
q/p is the equilibrium value, not the complete time-dependent solution.
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 First-order linear ODE do?
Solve y′+py=q with constant coefficients and an initial value.
How does the First-order linear ODE work?
The calculator applies y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0. The solution combines the equilibrium q/p with a transient exponential term that carries the initial condition.
What can I learn from the First-order linear 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.