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# how to solve DE

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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 $f(x).$

Aug 15, 2023

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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.

Aug 15, 2023