System of Differential Equations Calculator

Solve simultaneous linear differential equations: dx/dt = ax + by + f(t), dy/dt = cx + dy + g(t) by elimination (with forcing terms), and three equations X′ = AX by eigenvectors, with the constants from the starting values.

Equations	Two: x and y
dx/dt = a·x + b·y + f(t): a, b	1, 2
f(t) (optional)
dy/dt = c·x + d·y + g(t): c, d	3, 2
g(t) (optional)
x(0) (optional)
y(0) (optional)

Equations

dx/dt = a·x + b·y + f(t): a, b

f(t) (optional)

A forcing term, like e^(2t), t or sin(t). Leave empty for none.

dy/dt = c·x + d·y + g(t): c, d

g(t) (optional)

x(0) (optional)

y(0) (optional)

General solution

x = c1·e^(−t) + c2·e^(4t), y = (−2c1·e^(−t) + 3c2·e^(4t)) / 2

Eliminating one unknown gives a second-order equation whose auxiliary roots −1, 4 are the eigenvalues of [[1, 2], [3, 2]]. Both equations are checked by substituting the answer back.

Exact form. Not independently checked.

Method: Elimination (D-operator method)
Worked out with: AllCalcHub expression engine
Check: Every step that can be checked was checked

Eigenvalues: −1, 4
x(t): c1·e^(−t) + c2·e^(4t)
y(t): (−2c1·e^(−t) + 3c2·e^(4t)) / 2

About the System of Differential Equations Calculator

Differentiate the first equation and use the second to remove y: x″ − (a + d)x′ + (ad − bc)x = 0. Solve it like any constant-coefficient equation, then get y back from y = (x′ − ax) ÷ b. The auxiliary roots are the eigenvalues of the matrix [[a, b], [c, d]].

Real distinct roots give exponentials, a repeated root gives (c₁ + c₂t)e^(mt), and complex roots α ± iβ give e^(αt)(c₁ cos βt + c₂ sin βt).

The formula

x″ − (a + d)x′ + (ad − bc)x = 0, y = (x′ − ax)/b

Worked example

x′ = x + 2y, y′ = 3x + 2y

x″ − 3x′ − 4x = 0: m = 4, −1
x = c₁e^(4t) + c₂e^(−t), y = (3/2)c₁e^(4t) − c₂e^(−t)

Things to keep in mind

Forcing terms must be sums of polynomials, exponentials, sines and cosines. Three-equation systems need rational eigenvalues and have no forcing terms.

Questions people ask

Why are these the eigenvalues?

x = e^(mt)v solves the system exactly when Av = mv.

What does a complex root mean?

The solution spirals: oscillation with a growing or decaying amplitude, depending on the sign of α.