Simultaneous linear differential equations

If two or more dependent variables are functions of a single independent variable and their derivatives are involved, they are called simultaneous differential equations.

These equations commonly arise in engineering to model interacting dynamic systems such as electrical circuits, mechanical linkages, or control systems.

Example :

dx/dt + 4y = t
dy/dt + 2x = e'

The method of solving such equations is based on the process of elimination, similar to solving algebraic simultaneous equations.

General Form

A system of simultaneous linear differential equations can be written as:

• dx/dt = a₁₁x + a_12​y + f_1​(t)
• dy/dt = a₂₁x + a_22​y + f_2​(t)

Where:

• x, y are dependent variables,
• t is the independent variable,
• a_ij​ are constants or functions of t, and
• f₁(t), f₂(t) are known functions.

Methods of Solution

Elimination Method – For simple 2×2 systems.

• Eliminate one variable to reduce the system's higher-order differential equation.
• Solve for one variable, then find the other.

Matrix Method – Using determinants or eigenvalues for constant coefficients.

• Write the system as X^' = AX
• Solve using eigenvalues and eigenvectors of A.

Laplace Transform Method – For non-homogeneous equations and engineering applications.

• Transform the system into algebraic equations, solve, then apply the inverse Laplace transform.

Solved Question on Simultaneous Linear Differential Equations

Question 1: Solve the system dx/dt + 2y = t, dy/dt + x = e^t

Solution:

dx/dt = t - 2y , dy/dt = e^t - x

Differentiate the first equation:

d²x/dt² ​= 1 − 2dt/dy ( since = d/dt (t) = 1)

Differentiate first equation and substitute dy/dtdy/dtdy/dt:

x′′= 1 − 2(e^t −x) ⇒x′′− 2x =1−2e^t

• Homogeneous: x_h = C_1^{\sqrt{2t}} + C_2^{-\sqrt{2t}}
• Particular: xp=−1/2 + e^t

y = t - x' /2 = -\frac{\sqrt{2}}{2} C_1 e^{\sqrt{2} t} + \frac{\sqrt{2}}{2} C_2 e^{-\sqrt{2} t} + 2t - 21 e^t

Question 2: Solve dx/dt = y, dy/dt = - x

Solution:

x" = y' = - x

x" + x = 0

x = C₁cos t + C₂sin t

y = dx/dt = - C₁ sin t + C ₂ cos t

Question 3: Solve the system: dx/dt + y = e^t, dy/dt − x = 0

Solution:

Write it as :

x' = e^t - y, y' = x

Differentiate first :

x" = e^t - y' = e^t - x
x" + x = e^t

Homogenpus: x_h = C₁ cost + C₂ sint + 1/2

Particular A = 1/2
x = C₁ cost + C₂ sin t + 1/2 e^t

y = e^t - x' = C₁ sint - C₂ cost + 1/2e^t

x = C_1​cost + C₂​sint + 1/2e^t, y = C_1​sint − C₂​cost+1/2 ​e^t

Question 4: Solve and find constants if x(0) = 1, y(0) = 0: dx/dt = y , dy/dt = - x

Solution:

From Question 2:

x = C₁cos t + C₂sin t, y = - C₁ sin t + C ₂ cos t

At t = 0 :

x(0) = 1
C₁ = 1

y(0) = 0
C₂ = 1

x = cost , y = - sint

Unsolved Questions on Simultaneous Linear Differential Equations

Question 1: Solve dx​/dt = 2x + 3y, dy/dt ​= − 3x + 2y

Question 2: Solve using Laplace transform method: dx/dt ​+ 2y = e^2t, dy/dt ​− 2x = 0.

Question 3: Solve dx/dt ​+ 4y = sint, dy​/dt − 4x = cost.

Question 4: Solve the system of simultaneous differential equations: dx/dt − 3y = e^t(t+1), dy/dt +3x = e^t(2t).