The functions $f(x)$ and $g(x)$ satisfy $f(0) = 3,$ $g(0) = -4,$ and $f'(x) = 7f(x) + 2g(x)$ and $g'(x) = -4f(x) + g(x).$ Find $f(x).$

Question

The functions $f(x)$ and $g(x)$ satisfy $f(0) = 3,$ $g(0) = -4,$ and
\begin{align*}
f'(x) &= 7f(x) + 2g(x), \\
g'(x) &= -4f(x) + g(x).
\end{align*}Find $f(x).$
I'm not sure how to solve the following system of equations, could someone please help? I got the general form of $f(x)$ to be $(c_1+2c_2)e^{5x}-(c_1+c_2)e^{3x}$ for constants $c_1$ and $c_2$, but I'm not sure how to find these constants using the initial conditions. Thanks!

Answer 1

What you do is set up a system of equations by applying the initial conditions, as jDAQ mentioned. Before setting up the equations, I will just mention that you should also include $g(x):$ \begin{align*} f(x)&=(c_1+2c_2)e^{5x}-(c_1+c_2)e^{3x}\\ g(x)&=-(c_1+2c_2)e^{5x}+2(c_1+c_2)e^{3x}. \end{align*}

Here's how it would look: \begin{align*} f(0)&=3\\ g(0)&=-4\\ (c_1+2c_2)-(c_1+c_2)&=3\\ -(c_1+2c_2)+2(c_1+c_2)&=-4\\ c_2&=3\\ c_1&=-4. \end{align*} Then just re-write $f$ and $g$ with these values plugged in.