Math Problem Statement
Solve this question of Differential Equation till the end
Solution
We are tasked with solving the given second-order linear differential equation:
This is a non-homogeneous Cauchy-Euler equation. Let's solve it step by step.
Step 1: General Solution of the Homogeneous Equation
The associated homogeneous equation is:
We assume a solution of the form , and substitute into the homogeneous equation. Calculating derivatives:
Substitute these into the equation:
Simplify:
Factor out (since ):
Simplify the quadratic equation:
Factorize:
Thus, and .
The general solution of the homogeneous equation is: where and are arbitrary constants.
Step 2: Particular Solution of the Non-Homogeneous Equation
Now consider the full equation:
We seek a particular solution using the method of undetermined coefficients. Since the right-hand side is , we try:
Derivatives of :
Substitute and its derivatives into the equation:
[ x^2 \left[ A \left( 6x \ln x + 5x \right) + B \cdot 6x \right] + 2x \left[ A \left( 3x^2 \ln x + x^2 \right) + B \cdot 3x^2 \right] - 12 \left[ A x^3 \ln x + Bx^3 \right] = x^3 \ln x. ]
Simplify each term:
-
From :
-
From :
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From :
Combine all terms: [ \big[ 6A + 6A - 12A \big] x^3 \ln x + \big[ 5A + 2A + 6B + 6B - 12B \big] x^3 = x^3 \ln x. ]
Equating coefficients:
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For :
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For :
Thus, the particular solution is:
Step 3: General Solution
The general solution is the sum of the homogeneous and particular solutions:
Would you like detailed verification of any step? Here are 5 related questions for further exploration:
- How do we solve a Cauchy-Euler equation with different forms of non-homogeneous terms?
- What if the trial solution overlaps with the homogeneous solution?
- Why do we assume in solving Cauchy-Euler equations?
- How does the method of undetermined coefficients apply to non-polynomial terms?
- What are the applications of Cauchy-Euler equations in physics and engineering?
Tip: Always check if the form of the particular solution overlaps with the homogeneous solution. If it does, multiply the trial solution by to ensure linear independence.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Cauchy-Euler Equations
Method of Undetermined Coefficients
Homogeneous and Non-Homogeneous Solutions
Formulas
General Cauchy-Euler equation: x^2 d^2y/dx^2 + bx dy/dx + c y = f(x)
Quadratic characteristic equation: m^2 + bm + c = 0
Theorems
Cauchy-Euler Method
Superposition Principle for Linear Differential Equations
Suitable Grade Level
Undergraduate
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