Initial Value Problem Differential Equation

Question

Im am confused on how to do Initial Value Problems involved differential equations. Particularly, this one. $$ {{\rm dP}\left(t\right) \over {\rm d}t}=4\left({\rm e}^{t - 1} + t\right)\,, \qquad {\rm P}\left(1\right) = 20 $$

Answer 1

Integrate both sides:

$P(t) = 2t^2 + 4e^{t-1} + C $

$\rightarrow 2 + 4 + C = 20$

$\rightarrow C = 14$

$\therefore P(t) = 2t^2 + 4e^{t-1} + 14.$

Answer 2

Integrate both sides: $$P(t)=2t^2+4e^{t-1}+C$$ Use the condition $P(1)=20$: $$20=2*1^2+4e^{1-1}+C$$ $$20=2+4+C$$ $$C=14$$ So the final equation is: $$P(t)=2t^2+4e^{t-1}+14$$