6. Solve the equation x²y" – 4xy + 6y = 21x¬4 using variation of parameter method. - %3D

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Solve using Variation of parameter

 

**Problem 6: Differential Equation Using Variation of Parameters**

Solve the equation

\[ x^2 y'' - 4xy' + 6y = 21x^{-4} \]

using the variation of parameters method.

---

**Explanation for Educational Context:**

This problem involves solving a second-order linear differential equation using the variation of parameters method. Here, \( y'' \) denotes the second derivative of \( y \) with respect to \( x \), and \( y' \) is the first derivative of \( y \). The equation features variable coefficients, making variation of parameters an appropriate technique.

The given equation is 

\[ x^2 y'' - 4xy' + 6y = 21x^{-4}, \]

where \( y \) is the dependent variable, and \( x \) is the independent variable.

Key steps in the method of variation of parameters include:

1. **Finding the Complementary Solution ( \( y_c \) ):** Solve the associated homogeneous equation 

   \[ x^2 y'' - 4xy' + 6y = 0. \]

2. **Particular Solution ( \( y_p \) ):** Use the variation of parameters formula to find a particular solution to the non-homogeneous equation.

3. **General Solution ( \( y \) ):** Combine the complementary solution and the particular solution: 

   \[ y = y_c + y_p. \]

The variation of parameters involves calculating two functions, usually denoted \( u_1(x) \) and \( u_2(x) \), to construct the particular solution based on two linearly independent solutions of the homogeneous equation.
Transcribed Image Text:**Problem 6: Differential Equation Using Variation of Parameters** Solve the equation \[ x^2 y'' - 4xy' + 6y = 21x^{-4} \] using the variation of parameters method. --- **Explanation for Educational Context:** This problem involves solving a second-order linear differential equation using the variation of parameters method. Here, \( y'' \) denotes the second derivative of \( y \) with respect to \( x \), and \( y' \) is the first derivative of \( y \). The equation features variable coefficients, making variation of parameters an appropriate technique. The given equation is \[ x^2 y'' - 4xy' + 6y = 21x^{-4}, \] where \( y \) is the dependent variable, and \( x \) is the independent variable. Key steps in the method of variation of parameters include: 1. **Finding the Complementary Solution ( \( y_c \) ):** Solve the associated homogeneous equation \[ x^2 y'' - 4xy' + 6y = 0. \] 2. **Particular Solution ( \( y_p \) ):** Use the variation of parameters formula to find a particular solution to the non-homogeneous equation. 3. **General Solution ( \( y \) ):** Combine the complementary solution and the particular solution: \[ y = y_c + y_p. \] The variation of parameters involves calculating two functions, usually denoted \( u_1(x) \) and \( u_2(x) \), to construct the particular solution based on two linearly independent solutions of the homogeneous equation.
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