Transform the given differential equation into an equivalent system of first-order differential equations. x(³) = (x") ² - 3 cos (x') X Let x₁ = x, x₂ = x', and x3 =x". Complete the system below. ✔ X3 11 II 11
Transform the given differential equation into an equivalent system of first-order differential equations. x(³) = (x") ² - 3 cos (x') X Let x₁ = x, x₂ = x', and x3 =x". Complete the system below. ✔ X3 11 II 11
Chapter6: Exponential And Logarithmic Functions
Section6.7: Exponential And Logarithmic Models
Problem 4SE: Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of...
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![### Converting Higher-Order Differential Equations to Systems of First-Order Differential Equations
#### Objective:
Transform the given differential equation into an equivalent system of first-order differential equations.
#### Given Differential Equation:
\[ x^{(3)} = \left( x'' \right)^2 - 3 \cos \left( x' \right) \]
#### Steps to Transform:
To convert the third-order differential equation into a system of first-order differential equations, follow these instructions:
1. **Define new variables** to reduce the order of derivatives:
\[
x_1 = x, \quad x_2 = x', \quad x_3 = x''
\]
2. **Express the derivatives** of these new variables, leading to a system of first-order equations:
\[
x_1' = x'
\]
\[
x_2' = x''
\]
\[
x_3' = x'''
\]
3. **Substitute the given differential equation** into the defined variables:
\[
x_3' = \left( x_3 \right)^2 - 3 \cos \left( x_2 \right)
\]
#### Complete System of First-Order Differential Equations:
\[
\begin{cases}
x_1' = x_2 \\
x_2' = x_3 \\
x_3' = \left( x_3 \right)^2 - 3 \cos \left( x_2 \right)
\end{cases}
\]
This system now consists of three first-order differential equations equivalent to the original third-order differential equation provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F2b366ddb-43a0-4f84-a09a-7940aa9fe8b0%2Fo5mpmuf_processed.png&w=3840&q=75)
Transcribed Image Text:### Converting Higher-Order Differential Equations to Systems of First-Order Differential Equations
#### Objective:
Transform the given differential equation into an equivalent system of first-order differential equations.
#### Given Differential Equation:
\[ x^{(3)} = \left( x'' \right)^2 - 3 \cos \left( x' \right) \]
#### Steps to Transform:
To convert the third-order differential equation into a system of first-order differential equations, follow these instructions:
1. **Define new variables** to reduce the order of derivatives:
\[
x_1 = x, \quad x_2 = x', \quad x_3 = x''
\]
2. **Express the derivatives** of these new variables, leading to a system of first-order equations:
\[
x_1' = x'
\]
\[
x_2' = x''
\]
\[
x_3' = x'''
\]
3. **Substitute the given differential equation** into the defined variables:
\[
x_3' = \left( x_3 \right)^2 - 3 \cos \left( x_2 \right)
\]
#### Complete System of First-Order Differential Equations:
\[
\begin{cases}
x_1' = x_2 \\
x_2' = x_3 \\
x_3' = \left( x_3 \right)^2 - 3 \cos \left( x_2 \right)
\end{cases}
\]
This system now consists of three first-order differential equations equivalent to the original third-order differential equation provided.
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