3×3 Linear System ODE Solver

Published on: December 21, 2025

This 3×3 Linear System ODE Solver helps you solve a system of three first-order differential equations and shows each step clearly. It works with constant-coefficient homogeneous systems, rewrites the system in matrix form, computes eigenvalue information, uses the matrix exponential, and applies optional initial conditions to find a particular solution.

Step-by-step method

Write the three first-order ODEs clearly.
Isolate x′, y′, and z′ so each equation gives one derivative.
Rewrite the system in vector form X′ = A X.
Find the characteristic equation det(A − λI) = 0.
Find the eigenvalues and use them to understand the solution shape.
Use the matrix exponential formula X(t) = e^{At}C, or X(t)=e^{A(t−t0)}X0 when initial conditions are given.
Write the final component functions x(t), y(t), and z(t).

Formula bank:

Matrix form: X′ = A X

det(A − λI) = 0

X(t) = e^AtC

With X(t₀) = X₀: X(t) = e^A(t−t₀)X₀

Final answer is the three component functions x(t), y(t), z(t).

Example 1: dx/dt = x; dy/dt = 2y; dz/dt = -z, t, x, y, z, x(0)=1, y(0)=2, z(0)=3

Step 1 - Write the system.

In this problem: Start with the three first-order equations.

= x

= 2y

= −z

Step 2 - Isolate the three derivatives.

In this problem: Each equation must give one of x′, y′, or z′.

x′ = x

y′ = 2y

z′ = −z

Step 3 - Define the vector X.

In this problem: Collect the dependent variables into a column vector.

X = [

]

X′ = [

x′

y′

z′

]

Step 4 - Build the coefficient matrix.

In this problem: Read the coefficients of x, y, and z from the derivative equations.

A = [

1	0	0
0	2	0
0	0	-1

]

X′ = A X

Step 5 - Find the characteristic equation.

In this problem: Compute det(A − λI) = 0.

det(A − λI) = 0

−(1 + λ)(−1 + λ)(−2 + λ) = 0

Step 6 - List the eigenvalues.

In this problem: These control the growth, decay, oscillation, and stability of the system.

λ₁ = 1 (multiplicity 1)

λ₂ = 2 (multiplicity 1)

λ₃ = -1 (multiplicity 1)

Step 7 - Find eigenvectors when useful.

In this problem: Eigenvectors give the independent solution directions. If they are not convenient, the matrix exponential method still works.

λ = -1 has algebraic multiplicity 1

v₁ = [

]

λ = 1 has algebraic multiplicity 1

v₂ = [

]

λ = 2 has algebraic multiplicity 1

v₃ = [

]

Step 8 - Use the matrix exponential solution formula.

In this problem: For every constant-coefficient homogeneous system, the general solution is X(t)=e^{At}C.

X(t) = eAt C

C = [

]

Step 9 - Compute e^{At}.

In this problem: This matrix sends the initial vector or constant vector forward in time.

eAt = [

et	0	0
0	e2t	0
0	0	e−t

]

Step 10 - Multiply e^{At} by C.

In this problem: This gives the general solution before applying initial conditions.

x(t) = C1et

y(t) = C2e2t

z(t) = C3e−t

Step 11 - Write the initial vector.

In this problem: Collect x(t0), y(t0), z(t0) into X0.

t₀ = 0

X₀ = [

]

Step 12 - Use the IVP formula.

In this problem: Instead of solving for constants manually, use X(t)=e^{A(t−t0)}X0.

X(t) = eA(t−t₀)X₀

eA(t−t₀) = [

et	0	0
0	e2t	0
0	0	e−t

]

Step 13 - Multiply by the initial vector.

In this problem: This gives the particular solution satisfying all three initial conditions.

x(t) = et

y(t) = 2e2t

z(t) = 3e−t

Step 14 - Check stability.

In this problem: Use the real parts of the eigenvalues.

Mixed / saddle-type (some real parts > 0 and some < 0)

Step 15 - Final answer.

In this problem: Write the three component functions.

x(t) = et

y(t) = 2e2t

z(t) = 3e−t

Final answer: See final step

Example 2: dx/dt = y; dy/dt = -x; dz/dt = -2z, t, x, y, z, x(0)=1, y(0)=0, z(0)=3

Step 1 - Write the system.

In this problem: Start with the three first-order equations.

= y

= −x

= −2z

Step 2 - Isolate the three derivatives.

In this problem: Each equation must give one of x′, y′, or z′.

x′ = y

y′ = −x

z′ = −2z

Step 3 - Define the vector X.

In this problem: Collect the dependent variables into a column vector.

X = [

]

X′ = [

x′

y′

z′

]

Step 4 - Build the coefficient matrix.

In this problem: Read the coefficients of x, y, and z from the derivative equations.

A = [

0	1	0
-1	0	0
0	0	-2

]

X′ = A X

Step 5 - Find the characteristic equation.

In this problem: Compute det(A − λI) = 0.

det(A − λI) = 0

−(1 + λ2)(2 + λ) = 0

Step 6 - List the eigenvalues.

In this problem: These control the growth, decay, oscillation, and stability of the system.

λ₁ = −i (multiplicity 1)

λ₂ = i (multiplicity 1)

λ₃ = -2 (multiplicity 1)

Step 7 - Find eigenvectors when useful.

In this problem: Eigenvectors give the independent solution directions. If they are not convenient, the matrix exponential method still works.

λ = -2 has algebraic multiplicity 1

v₁ = [

]

λ = −i has algebraic multiplicity 1

v₂ = [

]

λ = i has algebraic multiplicity 1

v₃ = [

−i

]

Step 8 - Use the matrix exponential solution formula.

In this problem: For every constant-coefficient homogeneous system, the general solution is X(t)=e^{At}C.

X(t) = eAt C

C = [

]

Step 9 - Compute e^{At}.

In this problem: This matrix sends the initial vector or constant vector forward in time.

eAt = [

eit +

e−it

ie−it −

ieit

ieit −

ie−it

eit +

e−it

e−2t

]

Step 10 - Multiply e^{At} by C.

In this problem: This gives the general solution before applying initial conditions.

x(t) =

(C1(1 + e2it) + iC2(1 − e2it))e−it

y(t) =

(C2(1 + e2it) − iC1(1 − e2it))e−it

z(t) = C3e−2t

Step 11 - Write the initial vector.

In this problem: Collect x(t0), y(t0), z(t0) into X0.

t₀ = 0

X₀ = [

]

Step 12 - Use the IVP formula.

In this problem: Instead of solving for constants manually, use X(t)=e^{A(t−t0)}X0.

X(t) = eA(t−t₀)X₀

eA(t−t₀) = [

eit +

e−it

ie−it −

ieit

ieit −

ie−it

eit +

e−it

e−2t

]

Step 13 - Multiply by the initial vector.

In this problem: This gives the particular solution satisfying all three initial conditions.

x(t) = cos(t)

y(t) = −sin(t)

z(t) = 3e−2t

Step 14 - Check stability.

In this problem: Use the real parts of the eigenvalues.

Stable (some real parts < 0 and none > 0)

Step 15 - Final answer.

In this problem: Write the three component functions.

x(t) = cos(t)

y(t) = −sin(t)

z(t) = 3e−2t

Final answer: See final step