https://r2.erweima.ai/mathbot/image/1718381809113vi9eu2af.jpg

Let's proceed with solving problem (2) from the image.

### Problem (2)
\[
\begin{cases}
\frac{d^2x}{dt^2} + 2 \frac{dy}{dt} = 0, & x|_{t=0} = 0, \frac{dx}{dt}\bigg|_{t=0} = 1; \\
\frac{d^2y}{dt^2} - 2 \frac{dx}{dt} = 0, & y|_{t=0} = 1, \frac{dy}{dt}\bigg|_{t=0} = 1.
\end{cases}
\]

### Solution
1. **First Equation:**
   \[
   \frac{d^2x}{dt^2} = -2 \frac{dy}{dt}
   \]
2. **Second Equation:**
   \[
   \frac{d^2y}{dt^2} = 2 \frac{dx}{dt}
   \]

Take the derivative of the second equation with respect to \(t\):
\[
\frac{d^3y}{dt^3} = 2 \frac{d^2x}{dt^2}
\]

From the first equation, substitute \(\frac{d^2x}{dt^2} = -2 \frac{dy}{dt}\):
\[
\frac{d^3y}{dt^3} = 2(-2 \frac{dy}{dt}) = -4 \frac{dy}{dt}
\]

This is a third-order differential equation:
\[
\frac{d^3y}{dt^3} + 4 \frac{dy}{dt} = 0
\]

### Characteristic Equation
Assume a solution of the form \(y = e^{rt}\):
\[
r^3 + 4r = 0
\]
\[
r(r^2 + 4) = 0
\]
\[
r = 0, \pm 2i
\]

Thus, the general solution is:
\[
y(t) = C_1 + C_2 \cos(2t) + C_3 \sin(2t)
\]

### Initial Conditions for \(y(t)\)
\[
y(0) = 1 \implies C_1 + C_2 = 1
\]

\[
\frac{dy}{dt} = -2C_2 \sin(2t) + 2C_3 \cos(2t)
\]
\[
\frac{dy}{dt}\bigg|_{t=0} = 1 \implies 2C_3 = 1 \implies C_3 = \frac{1}{2}
\]

### Solution for \(x(t)\)
Since \(\frac{d^2x}{dt^2} = -2 \frac{dy}{dt}\):
\[
\frac{d^2x}{dt^2} = -2(-2C_2 \sin(2t) + 2C_3 \cos(2t))
\]
\[
\frac{d^2x}{dt^2} = 4C_2 \sin(2t) - 4C_3 \cos(2t)
\]

Integrate to find \(x(t)\):
\[
\frac{dx}{dt} = -2C_2 \cos(2t) - 2C_3 \sin(2t) + C_4
\]

Initial condition for \(\frac{dx}{dt}|_{t=0} = 1\):
\[
1 = -2C_2 + C_4
\]

Integrate again to find \(x(t)\):
\[
x(t) = -C_2 \sin(2t) + C_3 \cos(2t) + C_4 t + C_5
\]

Initial condition for \(x(0) = 0\):
\[
0 = C_5
\]

So, the solutions are:
\[
y(t) = C_1 + C_2 \cos(2t) + \frac{1}{2} \sin(2t)
\]
\[
x(t) = -C_2 \sin(2t) + \frac{1}{2} \cos(2t) + C

Explore the solution process for a system of second-order differential equations involving initial value problems. This detailed solution covers differential equations, initial conditions, and characteristic equations. Suitable for college-level students.

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