Question

(2t) + b exp(5t)" alt="y(t) = a exp(2t) + b exp(5t)" align="absmiddle" class="latex-formula">
$y(0) = 2$ and $y'(0) = 3$
Find a and b. I did the substitutions and everything and got $a= \frac{5}{3}$ and $b= \frac{-1}{3}$ but apparently this is not right

Answer 1

$y(0)=2\implies 2=ae^{2(0)}+be^{5(0)}\implies 2=a+b$

$y'(t)=2ae^{2t}+5be^{5t}$
$y'(0)=3\implies 3=2ae^{2(0)}+5be^{5(0)}\implies 3=2a+5b$

Solving this should yield $a=\dfrac73$ and $b=-\dfrac13$.