Questions: Show that the functions (x(t)=e^-t / 10 sin t) and (y(t)=frac110 e^-t / 10(-10 cos t+sin t)) are solutions of the initial value problem [ fracd xd t=-y fracd yd t=(1.01) x-(0.2) y ; x(0)=0, y(0)=-1 ]

$Show that the functions (x(t)=e^-t / 10 sin t) and (y(t)=frac110 e^-t / 10(-10 cos t+sin t)) are solutions of the initial value problem [ fracd xd t=-y fracd yd t=(1.01) x-(0.2) y ; x(0)=0, y(0)=-1 ]$

Transcript text: Show that the functions $x(t)=e^{-t / 10} \sin t$ and $y(t)=\frac{1}{10} e^{-t / 10}(-10 \cos t+\sin t)$ are solutions of the initial value problem \[ \begin{array}{l} \frac{d x}{d t}=-y \\ \frac{d y}{d t}=(1.01) x-(0.2) y ; x(0)=0, y(0)=-1 \end{array} \]

Solution

Solution Steps

To show that the given functions $ x(t) = e^{-t / 10} \sin t $ and $ y(t) = \frac{1}{10} e^{-t / 10}(-10 \cos t + \sin t) $ are solutions to the initial value problem, we need to:

Compute the derivatives $ \frac{dx}{dt} $ and $ \frac{dy}{dt} $.
Substitute $ x(t) $ and $ y(t) $ into the differential equations to verify that they satisfy both equations.
Check the initial conditions $ x(0) = 0 $ and $ y(0) = -1 $.

Step 1: Compute the Derivatives

First, we compute the derivatives of $ x(t) $ and $ y(t) $: \[ x(t) = e^{-t/10} \sin(t) \] \[ y(t) = 0.1 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} \] \[ \frac{dx}{dt} = -\frac{1}{10} e^{-t/10} \sin(t) + e^{-t/10} \cos(t) \] \[ \frac{dy}{dt} = -0.01 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} + 0.1 \left( 10 \sin(t) + \cos(t) \right) e^{-t/10} \]

Step 2: Verify the First Differential Equation

We substitute $ x(t) $ and $ y(t) $ into the first differential equation: \[ \frac{dx}{dt} = -y \] \[ -\frac{1}{10} e^{-t/10} \sin(t) + e^{-t/10} \cos(t) = -0.1 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} \] Both sides simplify to: \[ -\frac{1}{10} e^{-t/10} \sin(t) + e^{-t/10} \cos(t) \] Thus, the first differential equation is satisfied.

Step 3: Verify the Second Differential Equation

We substitute $ x(t) $ and $ y(t) $ into the second differential equation: \[ \frac{dy}{dt} = 1.01 x - 0.2 y \] \[ -0.01 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} + 0.1 \left( 10 \sin(t) + \cos(t) \right) e^{-t/10} = 1.01 e^{-t/10} \sin(t) - 0.02 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} \] Both sides simplify to: \[ -0.01 \left( \sin(t) - 10 \cos(t) \right) e^{-t/10} + 0.1 \left( 10 \sin(t) + \cos(t) \right) e^{-t/10} \] Thus, the second differential equation is satisfied.

Step 4: Check Initial Conditions

We check the initial conditions $ x(0) = 0 $ and $ y(0) = -1 $: \[ x(0) = e^{0} \sin(0) = 0 \] \[ y(0) = 0.1 \left( \sin(0) - 10 \cos(0) \right) e^{0} = 0.1 \left( 0 - 10 \right) = -1 \] Both initial conditions are satisfied.