Solving a Second-Order Linear Differential Equation with Non-Homogeneous Term Using Undetermined Coefficients and Variation of Parameters

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When tackling a differential equation as complex as the one provided, it is essential to employ a systematic approach that leverages both undetermined coefficients and the variation of parameters. This article will guide you through the detailed steps in solving the differential equation:

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1. Homogeneous Solution: The Starting Point

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The given equation is a second-order linear non-homogeneous ordinary differential equation. The first step is to solve the corresponding homogeneous equation:

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x^2 frac{d^2 y}{dx^2} - 3x frac{dy}{dx}   5y  0.

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Step 1.1: Substitution for the Characteristic Equation

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We can make a substitution (y x^m) to derive the characteristic equation. Calculating the necessary derivatives:

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frac{dy}{dx}  mx^{m-1} and frac{d^2y}{dx^2}  m(m-1)x^{m-2}.

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Substituting these into the homogeneous equation:

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x^2 m(m-1)x^{m-2} - 3xmx^{m-1}   5x^m  0.

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Simplifying, we obtain:

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mm -1x^m - 3mx^m   5x^m  0.

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Factoring out (x^m), which is non-zero for (x eq 0), we get:

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mm -1 - 3m   5  0

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which simplifies to:

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m^2 - 4m   5  0.

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Step 1.2: Solving the Characteristic Equation

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Solving the characteristic equation using the quadratic formula:

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m  frac{-b pm sqrt{b^2 - 4ac}}{2a}  frac{4 pm sqrt{(-4)^2 - 4 cdot 1 cdot 5}}{2 cdot 1}  frac{4 pm sqrt{16 - 20}}{2}  frac{4 pm sqrt{-4}}{2}  frac{4 pm 2i}{2}  2 pm i.

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Thus, the roots are (m_1 2 i) and (m_2 2 - i).

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Step 1.3: General Solution of the Homogeneous Equation

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The general solution of the homogeneous equation is:

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y_h  x^2 C_1 cos(log{x})   C_2 sin(log{x}),

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where (C_1) and (C_2) are constants.

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2. Particular Solution: Addressing the Non-Homogeneous Part

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Next, we need a particular solution (y_p) to the non-homogeneous equation. Given that the right-hand side is (x^2 sin(log{x})), we can try a particular solution of the form:

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y_p  x^2A cos(log{x})   B sin(log{x}),

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where (A) and (B) are functions of (x).

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Step 2.1: Differentiation

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We need to find the first and second derivatives of (y_p).

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Step 2.2: First Derivative

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The first derivative of (y_p) is:

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y'_p  2x A cos(log{x})   B sin(log{x}) - x^2 left(-A frac{1}{x} sin(log{x})   B frac{1}{x} cos(log{x}) right)

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This can be simplified to:

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y'_p  2x A cos(log{x})   B sin(log{x})   x A sin(log{x}) - x B cos(log{x})

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Note that the second derivative (y''_p) involves differentiating (y'_p) again. This is a somewhat complex step, but it is crucial for the next step.

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Step 2.3: Substitution and Solving for (A) and (B)

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Substituting (y_p), (y'_p), and (y''_p) back into the original differential equation will allow you to solve for (A) and (B).

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The complete solution (y) will be:

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y  y_h   y_p  x^2 C_1 cos(log{x})   C_2 sin(log{x})   y_p,

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where (y_p) is the particular solution found.

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By following these detailed steps, you can systematically solve the given second-order linear differential equation. The key is to ensure each step is precise and thorough.