2. Consider the equation t2a" + 2t.a' – 6x = t2. (1) Show that r1(t) = t² and r2(t) = t-3 are independent (a) solutions to the homogeneous equation t2a" + 2tx' – 6x 0. That is, show each is a solution and show the Wronskian is non-zero. (b) forget to make the coefficient on the a" term one before jumping in. Solve (1) using variation of parameters. Hint: Don't
2. Consider the equation t2a" + 2t.a' – 6x = t2. (1) Show that r1(t) = t² and r2(t) = t-3 are independent (a) solutions to the homogeneous equation t2a" + 2tx' – 6x 0. That is, show each is a solution and show the Wronskian is non-zero. (b) forget to make the coefficient on the a" term one before jumping in. Solve (1) using variation of parameters. Hint: Don't
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Differential Equation Problem
**2. Consider the equation**
\[ t^2 x'' + 2t x' - 6x = t^2. \tag{1} \]
**(a)** Show that \( x_1(t) = t^2 \) and \( x_2(t) = t^{-3} \) are independent solutions to the homogeneous equation
\[ t^2 x'' + 2t x' - 6x = 0. \]
That is, demonstrate that each is a solution and verify the Wronskian is non-zero.
**(b)** Solve (1) using the variation of parameters. **Hint:** Don’t forget to make the coefficient on the \( x'' \) term one before you start.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85346d8e-a543-42db-b008-9fe5a873d3b5%2F47b8c825-8a07-408a-910c-af6552e15dc3%2F342rx8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equation Problem
**2. Consider the equation**
\[ t^2 x'' + 2t x' - 6x = t^2. \tag{1} \]
**(a)** Show that \( x_1(t) = t^2 \) and \( x_2(t) = t^{-3} \) are independent solutions to the homogeneous equation
\[ t^2 x'' + 2t x' - 6x = 0. \]
That is, demonstrate that each is a solution and verify the Wronskian is non-zero.
**(b)** Solve (1) using the variation of parameters. **Hint:** Don’t forget to make the coefficient on the \( x'' \) term one before you start.
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