Answer
The actual result $74.691$
The approximation $66.425$
Difference: $74.691-66.425=8.266$
Work Step by Step
We are given $\frac{dy}{dx}=2e^{x}-y$
Using Euler's method
$g(x,y)=2e^{x}-y$
Since $x=0, y=100$
$g(x_{0},y_{0})=-98$ and
$y_{1}=y_{0}+g(x_{0},y_{0})h=100+(-98)\times0.1=90.2$
Now $x_{1}=0.1, y_{1}=90.2$
and $g(x_{1},y_{1})=-87.99$
Then $y_{2}=90.2+(-87.99)\times0.1=81.401$
$y_{3}=81.401+(-78.958)\times0.1=73.505$
$y_{4}=73.505+(-70.805)\times0.1=66.425$ $y(0.3)=y_{4}=66.425$
$\frac{dy}{dx}+y=2e^{x}$
$\int ydy=\int xdx$
This equation is written in the form
$\frac{dy}{dx}+P(x)y=Q(x)$
we can note that $P(x)=1$
The integrating factor is
$I(x)=e^{\int P(x)dx}=e^{\int dx}=e^{x}$
multiplying both sides of the differential equation by $e^{x}$ $e^{x}\frac{dy}{dx}+e^{x}y=2e^{2x}$
Write the terms on the left in the form $D_{x}[I(x)y]$
$D_{x}[e^{x}y]=2e^{2x}$
solve for y integrating both sides $e^{x}y=\int 2e^{2x}dx$
$e^{x}y=2\frac{e^{2x}}{2}+C$
$y=\frac{e^{2x}+C}{e^{x}}$
or $y=e^{x}+\frac{C}{e^{x}}$
Find the particular solution by substituting 0 for x and 100 for y $C=(y-e^{x})e^{x}=99$
$y=e^{x}+\frac{99}{e^{x}}$
$\rightarrow y=74.691$
Difference between the result and approximation: $74.691-66.425=8.266$
