Consider an undampened harmonic oscillator d'' + 7dy/dx + 10y = 0 with initial displacement 𝑦(0)=2 and initial velocity 𝑦′(0) = −13. Find displacement 𝑦(𝑡) at any time 𝑡

To find the displacement \( y(t) \) of an undamped harmonic oscillator described by the differential equation

\[ \frac{d^2y}{dt^2} + 7\frac{dy}{dt} + 10y = 0 \]

with initial conditions \( y(0) = 2 \) and \( y'(0) = -13 \), we proceed as follows:

1. **Solve the characteristic equation**: The characteristic equation for this differential equation is

   \[ r^2 + 7r + 10 = 0 \]

   Solving for \( r \):

   \[ r = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 10}}{2} \]
   \[ r = \frac{-7 \pm \sqrt{49 - 40}}{2} \]
   \[ r = \frac{-7 \pm \sqrt{9}}{2} \]
   \[ r = \frac{-7 \pm 3}{2} \]

   Therefore, the roots are:

   \[ r_1 = \frac{-7 + 3}{2} = -2 \]
   \[ r_2 = \frac{-7 - 3}{2} = -5 \]

2. **Write the general solution**: Given the roots are real and distinct, the general solution of the differential equation is:

   \[ y(t) = C_1 e^{-2t} + C_2 e^{-5t} \]

3. **Apply initial conditions to find constants**:

   - Use \( y(0) = 2 \):

     \[ y(0) = C_1 e^{0} + C_2 e^{0} = C_1 + C_2 = 2 \]

   - Use \( y'(0) = -13 \):

     First, compute \( y'(t) \):

     \[ y'(t) = -2C_1 e^{-2t} - 5C_2 e^{-5t} \]

     Then, at \( t = 0 \):

     \[ y'(0) = -2C_1 - 5C_2 = -13 \]

     So, we have the system of equations:

     \[
     \begin{cases}
     C_1 + C_2 = 2 \\
     -2C_1 - 5C_2 = -13
     \end{cases}
     \]

4. **Solve the system of equations**:

   From the first equation, we have:

   \[ C_2 = 2 - C_1 \]

   Substitute \( C_2 \) into the second equation:

   \[ -2C_1 - 5(2 - C_1) = -13 \]
   \[ -2C_1 - 10 + 5C_1 = -13 \]
   \[ 3C_1 - 10 = -13 \]
   \[ 3C_1 = -3 \]
   \[ C_1 = -1 \]

   Then, substitute \( C_1 \) back into \( C_2 = 2 - C_1 \):

   \[ C_2 = 2 - (-1) = 3 \]

5. **Write the particular solution**:

   \[ y(t) = -e^{-2t} + 3e^{-5t} \]

Therefore, the displacement \( y(t) \) at any time \( t \) is:

\[ y(t) = -e^{-2t} + 3e^{-5t} \]

Would you like more details or have any questions? Here are some more questions you might be interested in:

1. How do you solve a damped harmonic oscillator?
2. Can you explain the physical significance of the characteristic equation roots?
3. What are the differences between underdamped, critically damped, and overdamped oscillations?
4. How do you find the solution of a non-homogeneous differential equation?
5. Can you show the derivation of the characteristic equation from a second-order differential equation?

**Tip:** Always double-check your initial conditions and constants when solving differential equations to ensure your final solution is accurate.

Learn how to solve the undamped harmonic oscillator differential equation d'' + 7dy/dx + 10y = 0 with initial conditions y(0)=2 and y'(0)=-13. Step-by-step solution including finding characteristic roots, solving for constants, and deriving the particular solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation