Differential equations Solve the initial value problem - ( )Y,Y(0) - (-).5 4x(t)Note: enter 7e-5ty(t)Enter answers as a(t) and y(t), where Y(t):as 7e^(-5t).a(t)y(t)%3Dx(t) Preview:y(t) Preview:

The given differential equation can be written as: dY→dt=21−54Y→ and Y→0=−3−5let…

Answered: Solve the initial value problem dt %3D…

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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Differential equations

Solve the initial value problem - ( )Y,Y(0) - (-).
5 4
x(t)
Note: enter 7e-5t
y(t)
Enter answers as a(t) and y(t), where Y(t):
as 7e^(-5t).
a(t)
y(t)
%3D
x(t) Preview:
y(t) Preview:

Expert Solution

Step 1

The given differential equation can be written as:

$\begin{array}{l} \frac{d \vec{Y}}{d t} = [\begin{matrix} 2 & 1 \\ - 5 & 4 \end{matrix}] \vec{Y} and \vec{Y} (0) = [\begin{matrix} - 3 \\ - 5 \end{matrix}] \\ let \vec{Y} = [\begin{matrix} x (t) \\ y (t) \end{matrix}] \\ [\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} 2 & 1 \\ - 5 & 4 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] \\ x^{'} = 2 x + y and y^{'} = - 5 x + 4 y \end{array}$

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