Question

Solve the linear differential equation: (dy/dx) + 2y = e^(2x)

Answer 1

hmm dont you just solve the integrating factor and then do that thingy like \[y(I.F.) = \int Q(x)(I.F.) dx\]
thingy

Answer 2

you are familiar with that method right?

Answer 3

yes, i just can't get the exact answer. do you remember how to do it?

Answer 4

\[I.F. = e^{\int 2dx} \implies e^{2x}\]

that's a start

Answer 5

so then \[y(I.F.) = \int Q(x) (I.F.) dx\]
]\[\implies y(e^{2x}) = \int e^{2x} (e^{2x})dx\]
\[\implies y(e^{2x}) = \int e^{4x}dx\]
that should be straightforward now

Answer 6

Okay, so there's the form:

1/u(x) \[1\div u\left( x \right) \int\limits_{?}^{?}q( x) u(x)\]

Answer 7

so you got q(x)u(x)

Answer 8

or, as you put it, (I.F.)

Answer 9

i guess i did

Answer 10

so then under that integral, i get \[e^(4x) \div4\]

Answer 11

right \[\Large \frac{e^{4x}}{4}\]

Answer 12

and then that gets divided by (I.F.), so \[e^4x / e^2x \]

Answer 13

?

Answer 14

yup

Answer 15

dont forget the 4

Answer 16

\[\huge y = \frac{e^{4x}}{4e^{2x}}\]
simplify it to get the final answer

Answer 17

ah okay, the textbook answer simplified the e^4x and e^2x. I got it now. Thanks for walking me through

Answer 18

welcome ^_^