Solving a System of Differential Equations

The Problem

Part A - Find a solution of the homogeneous system with a t^α component:

dx/dt = (3/t)x − 4ty
dy/dt = (1/t³)x + (5/t)y

Part B - Using the solution you found, find all solutions of:

dx/dt = (3/t)x − 4ty + 5t³
dy/dt = (1/t³)x + (5/t)y − 4t

Solution Method

A system with variable coefficients. We find a homogeneous solution using a power-of-t ansatz, then use reduction of order (change of variables) to find the general solution of the non-homogeneous system.

Step 1: Homogeneous Solution - Power Ansatz

Find a solution of the homogeneous system whose components are powers of t

Guiding Question

Try a solution of the form x(t) = at^α, y(t) = bt^β

Substitute into the system and find α, β and the ratio a:b

Hint: from the equation structure α = β + 2. Try β = 3

What are the powers and the homogeneous solution?

x(t) = at^α, α = y(t) = bt^β, β =

Homogeneous solution: [a, b]^T = [ , ]^T

Enter integers (any scalar multiple is accepted)

🧐 Digital Friend - Step 1

Step 1 Solution

Ansatz:

x = at^α, y = bt^β with α = β + 2

The Homogeneous Solution

α = 5, β = 3, [a, b]^T = [−2, 1]^T
φ(t) = [−2t⁵, t³]^T

Step 2: Change of Variables

Using the homogeneous solution [−2t⁵, t³]^T, define:
x = ξ − 2t⁵η, y = t³η
Substitute into the non-homogeneous system and derive scalar equations

Guiding Question

After substitution, two scalar equations are obtained. What are the coefficients?

dξ/dt = (?/t)·ξ − 3t³
dη/dt = t^?·ξ − 4t⁻²

Hint: substitute x = ξ − 2t⁵η, y = t³η and simplify

What are the coefficients in the scalar equations?

dξ/dt = ( /t)·ξ − 3t³

dη/dt = t⁽ ⁾·ξ − 4t⁻²

🧐 Digital Friend - Step 2

Step 2 Solution

The substitution:

x = ξ − 2t⁵η, y = t³η

The Scalar Equations

dξ/dt = (5/t)ξ − 3t³
dη/dt = t⁻⁶ξ − 4t⁻²

Step 3: Solving the ξ Equation

Solve the first-order linear equation:
dξ/dt − (5/t)ξ = −3t³

Guiding Question

Find an integrating factor and solve. The solution has the form ξ = c₁t⁴ + At⁵

μ = e^{−∫(5/t)dt} = t⁻⁵

Hint: multiply by the integrating factor and integrate

What is the coefficient c₁ in ξ = c₁t⁴ + At⁵?

c₁ =

🧐 Digital Friend - Step 3

Step 3 Solution

The equation:

dξ/dt − (5/t)ξ = −3t³

Integrating factor:

μ = t⁻⁵

The Solution

ξ = 3t⁴ + At⁵

Step 4: Solving the η Equation

Substitute ξ = 3t⁴ + At⁵ into the equation:
dη/dt = t⁻⁶ξ − 4t⁻²

Guiding Question

Simplify and integrate. The solution has the form η = c₂t⁻¹ + A·ln(t) + B

dη/dt = −t⁻² + At⁻¹

Hint: integrate term by term

What is the coefficient c₂ of t⁻¹ in the solution?

c₂ =

🧐 Digital Friend - Step 4

Step 4 Solution

Substituting ξ:

dη/dt = t⁻⁶(3t⁴ + At⁵) − 4t⁻² = −t⁻² + At⁻¹

Integration:

η = t⁻¹ + A·ln(t) + B

The Solution

η = t⁻¹ + A·ln(t) + B

Step 5: The General Solution

Substitute back x = ξ − 2t⁵η, y = t³η
Find the particular solution (without constants A and B)

Guiding Question

Substitute ξ = 3t⁴ + At⁵ and η = t⁻¹ + A·ln(t) + B

x_p(t) = t^?, y_p(t) = t^?

Hint: the particular solution is obtained when A = B = 0

What are the powers in the particular solution?

x_p(t) = t⁽ ⁾ y_p(t) = t⁽ ⁾

🧐 Digital Friend - Step 5

Step 5 Solution

Back-substitution (A=B=0):

x = ξ − 2t⁵η = 3t⁴ − 2t⁵·t⁻¹ = 3t⁴ − 2t⁴ = t⁴

y component:

y = t³η = t³·t⁻¹ = t²

The Particular Solution

x_p(t) = t⁴, y_p(t) = t²

The Problem

Solution Method

Guiding Question

Step 1 Solution

The Homogeneous Solution

Guiding Question

Step 2 Solution

The Scalar Equations

Guiding Question

Step 3 Solution

The Solution

Guiding Question

Step 4 Solution

The Solution

Guiding Question

Step 5 Solution

The Particular Solution

The General Solution

Step 1 Solution - Homogeneous Solution via Power Ansatz

The Homogeneous Solution

Step 2 Solution - Change of Variables (Reduction of Order)

The Scalar Equations

Step 3 Solution - The ξ Equation

The Solution

Step 4 Solution - The η Equation

The Solution

Step 5 Solution - The General Solution

The General Solution