Question

y(x)=-1/14+12e^14x is this the solution to y'=14y+1

Answer 1

\[y'-14y=1\]
A few ways to solve this one... Easiest way (I think) is approaching this like any usual linear equation by finding the integrating factor.
\[\mu(x)=e^{\int(-14)~dt}=e^{-14x}\]
\[\begin{align*}e^{-14x}-14e^{-14x}y=e^{-14x}~~\iff~~y&=e^{14x}\int e^{-14x}~dx\\
&=e^{14x}\left(-\frac{1}{14}e^{-14x}+C\right)\\
&=-\frac{1}{14}+Ce^{14x}
\end{align*}\]
Do you have an initial condition to find \(C\)?