Euler - Operator Splitting for $y'+y=3e^2t$, $y(0)=1$

Question

$y'+y=3e^2t$, $y(0)=1$

How can we solve this equation with operator splitting methods in matlab by using forward Euler method?

Answer 1

$${ y }^{ \prime }+y=3{ e }^{ 2t }\\ \\ { y }^{ \prime }+y=0\\ { y }^{ \prime }=-y\\ \int { \frac { dy }{ y } } =-\int { dt } \\ \ln { y } =-t+C\\ y=C{ e }^{ -t }\\ y=C\left( t \right) { e }^{ -t }\\ { y }^{ \prime }={ C }^{ \prime }\left( t \right) { e }^{ -t }-C\left( t \right) { e }^{ -t }\\ { C }^{ \prime }\left( t \right) { e }^{ -t }-C\left( t \right) { e }^{ -t }+C\left( t \right) { e }^{ -t }=3{ e }^{ 2t }\\ { C }^{ \prime }\left( t \right) { e }^{ -t }=3{ e }^{ 2t }\\ { C }^{ \prime }\left( t \right) =3{ e }^{ 3t }\\ C\left( t \right) =\int { 3{ e }^{ 3t } } dt={ e }^{ 3t }+{ C }_{ 1 }\\ y={ e }^{ -t }\left( { e }^{ 3t }+{ C }_{ 1 } \right) $$ and consider the condition that $y\left( 0 \right) =1$ $$y\left( 0 \right) =1+C=1\\ C=0$$ so the final answer is

$$y={ e }^{ 2t } $$