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.
Understand First-order linear ODE
One idea, three depths
Choose how deeply to explain First-order linear ODE
First-order linear ODE: Solve y′+py=q with constant coefficients and an initial value.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using First-order linear ODE to answer this question: solve y′+py=q with constant coefficients and an initial value? Enter Initial value y₀, Coefficient p, Constant q, and 1 other input; the calculator shows y(t). For example: For y′+2y=10, y(0)=1, the solution at t=1 is about 4.46. The answer tells you y(t).
Age 15Explain it to a 15-year-oldConnect it to the formula
The solution combines the equilibrium q/p with a transient exponential term that carries the initial condition. The rule is y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0. Its input values are Initial value y₀, Coefficient p, Constant q, Time t, and the main result is y(t). For example: For y′+2y=10, y(0)=1, the solution at t=1 is about 4.46.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated first-order linear ode relation over the valid real-number domain stated below. The implemented relation is y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0, evaluated from Initial value y₀, Coefficient p, Constant q, Time t to produce y(t). The solution combines the equilibrium q/p with a transient exponential term that carries the initial condition. q/p is the equilibrium value, not the complete time-dependent solution.
Inputs and valid domain
- Initial value y₀ must be a finite real number.
- Coefficient p must be a finite real number.
- Constant q must be a finite real number.
- Time t must be a finite real number.
Important boundary: q/p is the equilibrium value, not the complete time-dependent solution.
The formula
y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0
How the calculator works through it
It substitutes Initial value y₀, Coefficient p, Constant q, Time t into the formula and exposes every numerical step above. The main output is y(t), accompanied by Equilibrium q/p, Transient term.
Read the result correctly
The y(t) is the direct answer to “solve y′+py=q with constant coefficients and an initial value.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For y′+2y=10, y(0)=1, the solution at t=1 is about 4.46.
Where this model stops being reliable
q/p is the equilibrium value, not the complete time-dependent solution.
Learn it by changing one value
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.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why First-order linear ODE works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
First-order linear ODE uses y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives and changing systems
A derivative describes the changing quantity that First-order linear ODE models or approximates.
Review this foundation about 7 min
Optional enrichment
- Exponential solution behaviour
Exponential behaviour helps you recognise common growth, decay and response patterns related to First-order linear ODE.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Initial value y₀, Coefficient p, Constant q, Time t.
- Evaluate the principal relationship: y(t)=q/p+(y₀−q/p)e⁻ᵖᵗ, p≠0.
- Return y(t) and check the domain conditions described above.
Python
from math import *
def linear_first_order_ode(a, b, c, x) -> float:
return ((c / b) + ((a - (c / b)) * exp((-(b * x)))))
assert abs(linear_first_order_ode(1, 2, 10, 1) - 4.458658867053549) < 1e-6 * max(1.0, abs(4.458658867053549))
C
#include <assert.h>
#include <math.h>
double linear_first_order_ode(double a, double b, double c, double x) {
return ((c / b) + ((a - (c / b)) * exp((-(b * x)))));
}
int main(void) {
const double expected = 4.458658867053549;
const double actual = linear_first_order_ode(1, 2, 10, 1);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double linear_first_order_ode(double a, double b, double c, double x) {
return ((c / b) + ((a - (c / b)) * std::exp((-(b * x)))));
}
int main() {
constexpr double expected = 4.458658867053549;
const double actual = linear_first_order_ode(1, 2, 10, 1);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double linear_first_order_ode(double a, double b, double c, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern exp
global linear_first_order_ode
section .text
linear_first_order_ode:
push rbp
mov rbp, rsp
sub rsp, 96
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd xmm0, [rbp-24]
divsd xmm0, [rbp-16]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-24]
divsd xmm0, [rbp-16]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-8]
subsd xmm0, [rbp-72]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-32]
movsd [rbp-96], xmm0
pxor xmm0, xmm0
subsd xmm0, [rbp-96]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-88]
call exp wrt ..plt
movsd [rbp-80], xmm0
movsd xmm0, [rbp-64]
mulsd xmm0, [rbp-80]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
addsd xmm0, [rbp-56]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
leave
ret
MATLAB
function result = linear_first_order_ode(a, b, c, x)
result = ((c / b) + ((a - (c / b)) * exp((-(b * x)))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_, x_] := ((c / b) + ((a - (c / b)) * Exp[(-(b * x))]));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). First-Order Linear ODE Calculator. MW SysArc Tools. https://math.mwsysarc.com/differential-equations/first-order-linear-constant
MLA 9
MW SysArc. “First-Order Linear ODE Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/differential-equations/first-order-linear-constant. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “First-Order Linear ODE Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/differential-equations/first-order-linear-constant.
Harvard
MW SysArc (2026) ‘First-Order Linear ODE Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/differential-equations/first-order-linear-constant (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_linear_first_order_ode_2026,
author = {{MW SysArc}},
title = {First-Order Linear ODE Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/differential-equations/first-order-linear-constant},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - First-Order Linear ODE Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/differential-equations/first-order-linear-constant
N1 - Published July 21, 2026
ER -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 . Calculations tested .