Question

Find the solution to dy/dt = 7 y satisfying y(9) = 5

Answer 1

$\rm \dfrac{dy}{dt}=7y,\qquad\qquad\qquad y(9)=5$

I'm not sure what methods you've learned up to this point but one option is to apply separation of variables:

$\rm \dfrac{dy}{y}=7dt$

and then integrate from there,

$\rm ln|y|=7t+c$

exponentiate to isolate y,

$\rm |y|=e^{7t+c}$

Apply exponent rule,

$\rm |y|=e^c e^{7t}$

rename this e^c as some new constant, perhaps A,

$\rm \pm y=A e^{7t}\qquad\qquad A>0$

This A can only be positive, non-zero, but absorbing the plus/minus fixes that restriction,

$\rm y=A e^{7t}\qquad\qquad A\ne0$

Use your initial information to solve for this unknown value A,

$\rm y(9)=A e^{7\cdot9}=5$

solving for A, dividing by the exponential,

$\rm A=5e^{-63}$

So we get a final result of

$\rm y(t)=5e^{-63}e^{7t}$

apply exponent rule again to get a better looking answer, and factor,

$\rm y(t)=5e^{7(t-9)}$

Lemme know if too confusing.