Apply the Runge-Kutta 2nd order method (by hand) to the following equation with a step size of h = 0.15 s. Find values for y(t) and y'(t) at t = 0.15 s and t = 0.3 s. Be sure to show your calculations in detail. %3D 100y" + 700 – 1200y = 250t with y(0) = 0.5 and y'(0) = 2
Apply the Runge-Kutta 2nd order method (by hand) to the following equation with a step size of h = 0.15 s. Find values for y(t) and y'(t) at t = 0.15 s and t = 0.3 s. Be sure to show your calculations in detail. %3D 100y" + 700 – 1200y = 250t with y(0) = 0.5 and y'(0) = 2
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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![**Applying the Runge-Kutta 2nd Order Method**
To solve the differential equation using the Runge-Kutta 2nd order method, we consider:
\[ 100y'' + 700 - 1200y = 250t \]
Given initial conditions:
\[ y(0) = 0.5, \quad y'(0) = 2 \]
**Step Size**: \( h = 0.1 \, s \)
**Objective**: Find values for \( y(t) \) and \( y'(t) \) at \( t = 0.15 \, s \) and \( t = 0.3 \, s \).
**Table Setup**:
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
t & k_{1,z_1} & k_{1,z_2} & k_{2,z_1} & k_{2,z_2} & z_1 & z_2 \\
\hline
0 & - & - & - & - & 0.5 & 2 \\
\hline
0.1 & & & & & & \\
\hline
0.2 & & & & & & \\
\hline
\end{array}
\]
To compute the entries \( k_{1,z_1}, k_{1,z_2}, k_{2,z_1}, k_{2,z_2} \), follow the Runge-Kutta 2nd order steps:
1. Calculate the intermediary slopes \( k_{1,z_1} \), \( k_{1,z_2} \) at each \( t \).
2. Use these to get \( k_{2,z_1} \) and \( k_{2,z_2} \) for the next step.
3. Finally, compute \( z_1 \), \( z_2 \) at \( t = 0.15 \, s \) and \( t = 0.3 \, s \).
**Detailed Calculation Process**: Use these values to find the new estimates for \( y(t) \) and \( y'(t) \) iteratively by applying the formulae associated with the Runge-Kutta method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c68f49e-68dd-4957-98f9-2eef30804977%2F4b63e0ec-b6a3-48eb-9a8d-50a112620f89%2Fdkavpy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Applying the Runge-Kutta 2nd Order Method**
To solve the differential equation using the Runge-Kutta 2nd order method, we consider:
\[ 100y'' + 700 - 1200y = 250t \]
Given initial conditions:
\[ y(0) = 0.5, \quad y'(0) = 2 \]
**Step Size**: \( h = 0.1 \, s \)
**Objective**: Find values for \( y(t) \) and \( y'(t) \) at \( t = 0.15 \, s \) and \( t = 0.3 \, s \).
**Table Setup**:
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
t & k_{1,z_1} & k_{1,z_2} & k_{2,z_1} & k_{2,z_2} & z_1 & z_2 \\
\hline
0 & - & - & - & - & 0.5 & 2 \\
\hline
0.1 & & & & & & \\
\hline
0.2 & & & & & & \\
\hline
\end{array}
\]
To compute the entries \( k_{1,z_1}, k_{1,z_2}, k_{2,z_1}, k_{2,z_2} \), follow the Runge-Kutta 2nd order steps:
1. Calculate the intermediary slopes \( k_{1,z_1} \), \( k_{1,z_2} \) at each \( t \).
2. Use these to get \( k_{2,z_1} \) and \( k_{2,z_2} \) for the next step.
3. Finally, compute \( z_1 \), \( z_2 \) at \( t = 0.15 \, s \) and \( t = 0.3 \, s \).
**Detailed Calculation Process**: Use these values to find the new estimates for \( y(t) \) and \( y'(t) \) iteratively by applying the formulae associated with the Runge-Kutta method.
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