Recast the system of higher order differential equations as a first order system. u""=2u-3v v"=3tu'v'- cos u
Recast the system of higher order differential equations as a first order system. u""=2u-3v v"=3tu'v'- cos u
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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![## Converting Higher Order Differential Equations to a First Order System
In many mathematical and scientific contexts, it is useful to recast higher order differential equations as a system of first order equations. This is because first order systems are often easier to analyze and can be solved using a wide range of numerical methods.
Consider the following system of higher order differential equations:
\[ u''' = 2u - 3v \]
\[ v'' = 3tu'v' - \cos u \]
To convert this system into a first order form, we need to introduce new variables to represent each derivative. Let:
\[ x_1 = u \]
\[ x_2 = u' \]
\[ x_3 = u'' \]
\[ x_4 = v \]
\[ x_5 = v' \]
With these substitutions, the original higher order system can be expressed in terms of first order differential equations:
From \( u''' = 2u - 3v \):
\[ x_1' = x_2 \]
\[ x_2' = x_3 \]
\[ x_3' = 2x_1 - 3x_4 \]
From \( v'' = 3tu'v' - \cos u \):
\[ x_4' = x_5 \]
\[ x_5' = 3tx_2x_5 - \cos x_1 \]
Thus, the original higher order system is recast as the following first order system:
\[ \frac{dx_1}{dt} = x_2 \]
\[ \frac{dx_2}{dt} = x_3 \]
\[ \frac{dx_3}{dt} = 2x_1 - 3x_4 \]
\[ \frac{dx_4}{dt} = x_5 \]
\[ \frac{dx_5}{dt} = 3tx_2x_5 - \cos x_1 \]
This system is now a first order differential system, which can be solved using standard methods for solving first order systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8aa10308-c186-4690-945e-554747bf90d8%2Fed9d416c-9375-42a2-b887-9d415debaa35%2Fu4opu91_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Converting Higher Order Differential Equations to a First Order System
In many mathematical and scientific contexts, it is useful to recast higher order differential equations as a system of first order equations. This is because first order systems are often easier to analyze and can be solved using a wide range of numerical methods.
Consider the following system of higher order differential equations:
\[ u''' = 2u - 3v \]
\[ v'' = 3tu'v' - \cos u \]
To convert this system into a first order form, we need to introduce new variables to represent each derivative. Let:
\[ x_1 = u \]
\[ x_2 = u' \]
\[ x_3 = u'' \]
\[ x_4 = v \]
\[ x_5 = v' \]
With these substitutions, the original higher order system can be expressed in terms of first order differential equations:
From \( u''' = 2u - 3v \):
\[ x_1' = x_2 \]
\[ x_2' = x_3 \]
\[ x_3' = 2x_1 - 3x_4 \]
From \( v'' = 3tu'v' - \cos u \):
\[ x_4' = x_5 \]
\[ x_5' = 3tx_2x_5 - \cos x_1 \]
Thus, the original higher order system is recast as the following first order system:
\[ \frac{dx_1}{dt} = x_2 \]
\[ \frac{dx_2}{dt} = x_3 \]
\[ \frac{dx_3}{dt} = 2x_1 - 3x_4 \]
\[ \frac{dx_4}{dt} = x_5 \]
\[ \frac{dx_5}{dt} = 3tx_2x_5 - \cos x_1 \]
This system is now a first order differential system, which can be solved using standard methods for solving first order systems.
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