ty" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = ť". a) Verify that y(t) = t is a solution of t²y" + bty' + cy = 0 if r is a solution r²+ (b-1)r + c = 0.
ty" + bty + cy=0 are called Euler's equation. For this type of differential equations, we look for solutions of the type y(t) = ť". a) Verify that y(t) = t is a solution of t²y" + bty' + cy = 0 if r is a solution r²+ (b-1)r + c = 0.
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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![### Euler's Differential Equation
The differential equation of the form:
\[ t^2 y'' + bty' + cy = 0 \]
is called Euler's equation.
For this type of differential equation, we look for solutions of the type \( y(t) = t^r \).
#### Problem
Verify that \( y(t) = t^r \) is a solution of \( t^2 y'' + bty' + cy = 0 \) if \( r \) is a solution of the characteristic equation:
\[ r^2 + (b - 1)r + c = 0 \]
#### Explanation
a) To verify the solution, we start by assuming the solution \( y(t) = t^r \). We then compute the necessary derivatives and substitute them into the given differential equation. Through this process, if the original differential equation holds true, we also derive the characteristic equation \( r^2 + (b - 1)r + c = 0 \), confirming the correctness of the solution.
This method is used to solve Euler's equations by transforming them into algebraic equations, which are easier to handle and solve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47a605b-1167-4b41-bc40-883b330bf596%2Ffdf2e2b5-f551-4908-9089-7a1e07929971%2Fp8kc6bn_processed.png&w=3840&q=75)
Transcribed Image Text:### Euler's Differential Equation
The differential equation of the form:
\[ t^2 y'' + bty' + cy = 0 \]
is called Euler's equation.
For this type of differential equation, we look for solutions of the type \( y(t) = t^r \).
#### Problem
Verify that \( y(t) = t^r \) is a solution of \( t^2 y'' + bty' + cy = 0 \) if \( r \) is a solution of the characteristic equation:
\[ r^2 + (b - 1)r + c = 0 \]
#### Explanation
a) To verify the solution, we start by assuming the solution \( y(t) = t^r \). We then compute the necessary derivatives and substitute them into the given differential equation. Through this process, if the original differential equation holds true, we also derive the characteristic equation \( r^2 + (b - 1)r + c = 0 \), confirming the correctness of the solution.
This method is used to solve Euler's equations by transforming them into algebraic equations, which are easier to handle and solve.
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