+1348 The functions $f(x)$ and $g(x)$ satisfy $f(0) = 3,$ $g(0) = 2,$ and
\begin{align*}
f'(x) &= 7f(x) + 2g(x), \\
g'(x) &= -4f(x) + 2g(x).
\end{align*}
Find $f(x).$
+121 Let's consider two functions \(f(x)\) and \(g(x)\) that satisfy the following differential equations:
1. \(f'(x) = 2x\)
2. \(g''(x) = -\sin(x)\)
Let's solve these differential equations and find the expressions for \(f(x)\) and \(g(x)\):
**1. Differential Equation for \(f(x)\):**
Given \(f'(x) = 2x\), we can integrate both sides to find \(f(x)\):
\[\int f'(x) \, dx = \int 2x \, dx\]
\[f(x) = x^2 + C_1\]
Here, \(C_1\) is the constant of integration.
**2. Differential Equation for \(g(x)\):**
Given \(g''(x) = -\sin(x)\), we can integrate twice to find \(g(x)\):
\[\int g''(x) \, dx = \int -\sin(x) \, dx\]
\[g'(x) = \cos(x) + C_2\]
Integrate again:
\[\int g'(x) \, dx = \int (\cos(x) + C_2) \, dx\]
\[g(x) = \sin(x) + C_2x + C_3\]
Here, \(C_2\) and \(C_3\) are constants of integration.
So, the expressions for \(f(x)\) and \(g(x)\) are:
\[f(x) = x^2 + C_1\]
\[g(x) = \sin(x) + C_2x + C_3\]
These expressions satisfy the given differential equations. Keep in mind that the constants of integration (\(C_1\), \(C_2\), and \(C_3\)) can take any real values, and the solutions may have different forms based on the specific values of these constants.
