Just Number 36 ### Initial Value Problems with Cauchy-Euler Equations**Objective**: Find the general solution of the following differential equations, and then solve the given initial value problem for \( t \geq 1 \).1. **Equation 33**: \[ t^2y'' + 2ty' - y = 0; \quad y(1) = 2, \; y'(1) = 0 \]2. **Equation 34**: \[ t^2y'' + 2ty' = 0; \quad y(1) = 0, \; y'(1) = 6 \]3. **Equation 35**: \[ t^2y'' - 2y = 0; \quad y(1) = 5, \; y'(1) = -3 \]4. **Equation 36**: \[ t^2y'' + 4y' - 4y = 0; \quad y(1) = 5, \; y'(1) = -3 \]5. **Equation 37**: \[ t^2y'' + 6y' = 0; \quad y(1) = 8, \; y'(1) = -12 \]Each equation is a second-order linear homogeneous differential equation of the form typically encountered in Cauchy-Euler problems. The challenge involves finding both the general solution of the differential equation and solving for particular solutions using given initial conditions. This process often utilizes transformations or characteristic equations to simplify the problem into a more manageable form.

Name: Just Number 36 ### Initial Value Problems with Cauchy-Euler Equations*
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: Just Number 36 ### Initial Value Problems with Cauchy-Euler Equations**Objective**: Find the general solution of the following differential equations, and then solve the given initial value problem for \( t \geq 1 \).1. **Equation 33**: \[ t^2y''