confusion about differential eq question

Question

i was using sympy to solve Differential eqs this was the code enter image description here

what does that C mean in the output and also is the answer right?Im new to solving D.E and im actually clueless where to start with

Answer 1

The C1 is an arbitrary constant. It appears during the integration. Let's try to solve the equation. ($f(x) = y$ for simplicity)

$-5y+y'=2$

$\frac{dy}{dx}=2+5y$

$\frac{1}{2+5y}dy=dx$

Then we integrate both sides. Notice that an arbitrary constant will appear since the integration is not definite.

$\frac{1}{5}ln(5y+2)=x+C$

$5y+2=e^{5x+C}=e^{5x}e^C$

Since $e^C$ is a constant, we can substitute it with an arbitrary constant which is $C1$. You can try to solve for $y$ and you will get the result.