Question

Why isn't \[y(x)=e^{9x}\] the general solution of \[y'=9y\]?

Answer 1

\[\Large y = e^{9x}\]
\[\Large y \ ' = 9e^{9x}\]

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\[\Large y \ ' = 9y\]
\[\Large 9e^{9x} = 9(e^{9x})\]
\[\Large 9e^{9x} = 9e^{9x} \ \ {\LARGE \color{green}{\checkmark}}\]

It's definitely a solution, but is it the only solution? Are there others?

Answer 2

@jim_thompson5910, nicely done!!

Answer 3

thanks @wmj259

Answer 4

the way to find the general solution would be to do this


\[\Large y \ ' = 9y\]
\[\Large \frac{dy}{dx} = 9y\]
\[\Large \frac{dy}{y} = 9dx\]
\[\Large \int\frac{dy}{y} = \int9dx\]
\[\Large \ln(|y|) = 9x+C\]
\[\Large |y| = e^{9x+C}\]
\[\Large |y| = e^{9x}*e^C\]
\[\Large |y| = e^C*e^{9x}\]
\[\Large y = \pm e^C*e^{9x}\]
\[\Large y = A*e^{9x}\]

where \(\Large A = \pm e^C\). So that is the general solution which encompasses every solution

Answer 5

I got it =) once I saw your first comment I remembered there is an unknown C in 'general solutions. thanks.

Answer 6

you're welcome