Find a general solution to the given Cauchy–Euler equation for t> 0. d²y dy 3t2- 15y = 0 dt dt2
Find a general solution to the given Cauchy–Euler equation for t> 0. d²y dy 3t2- 15y = 0 dt dt2
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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![**Solving Initial Value Problems for the Cauchy-Euler Equation**
In this example, we aim to solve the initial value problem for the Cauchy-Euler equation provided below:
\[ t^2 y''(t) + 10t y'(t) + 20y(t) = 0 \]
\[ y(1) = -1, \quad y'(1) = 7 \]
To solve the problem, follow these steps:
1. Begin by solving the homogeneous Cauchy-Euler differential equation:
\[ t^2 y''(t) + 10t y'(t) + 20y(t) = 0 \]
2. Next, determine the general solution for the differential equation.
3. Use the provided initial conditions to find the particular solution.
Initial conditions:
\[ y(1) = -1 \]
\[ y'(1) = 7 \]
4. Substitute the initial conditions into the general solution to solve for any constants.
Finally, write the particular solution of \( y(t) \).
\[ \text{The solution is} \ y(t) = \boxed{\phantom{y(t)}}. \]
_exclamation_circle_ **Note for Students:** This is a complete example of how to approach solving a second-order linear differential equation with variable coefficients. The Cauchy-Euler equation is a common type of such equations, frequently encountered in engineering and physics problems. Follow each of these steps to arrive at the specific solution for your problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffae14c32-d1e2-4020-8be9-d90c8f175447%2F711325ce-0e8a-4257-818d-bfd7100d40c5%2Fnb26za9.png&w=3840&q=75)
Transcribed Image Text:**Solving Initial Value Problems for the Cauchy-Euler Equation**
In this example, we aim to solve the initial value problem for the Cauchy-Euler equation provided below:
\[ t^2 y''(t) + 10t y'(t) + 20y(t) = 0 \]
\[ y(1) = -1, \quad y'(1) = 7 \]
To solve the problem, follow these steps:
1. Begin by solving the homogeneous Cauchy-Euler differential equation:
\[ t^2 y''(t) + 10t y'(t) + 20y(t) = 0 \]
2. Next, determine the general solution for the differential equation.
3. Use the provided initial conditions to find the particular solution.
Initial conditions:
\[ y(1) = -1 \]
\[ y'(1) = 7 \]
4. Substitute the initial conditions into the general solution to solve for any constants.
Finally, write the particular solution of \( y(t) \).
\[ \text{The solution is} \ y(t) = \boxed{\phantom{y(t)}}. \]
_exclamation_circle_ **Note for Students:** This is a complete example of how to approach solving a second-order linear differential equation with variable coefficients. The Cauchy-Euler equation is a common type of such equations, frequently encountered in engineering and physics problems. Follow each of these steps to arrive at the specific solution for your problem.
![### Cauchy–Euler Equation Solution
#### Problem Statement
Find a general solution to the given Cauchy–Euler equation for \( t > 0 \).
\[ t^2 \frac{d^2 y}{dt^2} + 3t \frac{dy}{dt} - 15y = 0 \]
#### Solution
Upon solving the differential equation, the general solution can be expressed in the form:
\[ y(t) = \boxed{e} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffae14c32-d1e2-4020-8be9-d90c8f175447%2F711325ce-0e8a-4257-818d-bfd7100d40c5%2Fwhjed6zx.png&w=3840&q=75)
Transcribed Image Text:### Cauchy–Euler Equation Solution
#### Problem Statement
Find a general solution to the given Cauchy–Euler equation for \( t > 0 \).
\[ t^2 \frac{d^2 y}{dt^2} + 3t \frac{dy}{dt} - 15y = 0 \]
#### Solution
Upon solving the differential equation, the general solution can be expressed in the form:
\[ y(t) = \boxed{e} \]
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