`y' + 3y = e^(3x)` Solve the first-order differential equation

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Given` y'+3y=e^(3x)`

when the first order linear ordinary differential equation has the form of

`y'+p(x)y=q(x)`

then the general solution is ,

`y(x)=((int e^(int p(x) dx) *q(x)) dx +c)/e^(int p(x) dx)`

so,

`y'+3y=e^(3x)--------(1)`

`y'+p(x)y=q(x)---------(2)`

on comparing both we get,

`p(x) = 3 and q(x)=e^(3x)`

so on solving with the above general solution we get:

y(x)=`((int e^(int p(x) dx) *q(x)) dx +c)/e^(int p(x) dx)`

=`((int e^(int 3 dx) *(e^(3x))) dx +c)/e^(int 3 dx)`

first we shall solve

`e^(int 3 dx)=e^(3x) `     

so

proceeding further, we get

y(x) =`((int e^(int 3 dx) *(e^(3x))) dx +c)/e^(int 3 dx)`

=`((int e^(3x) *(e^(3x))) dx +c)/e^(3x)`

=`((int e^(6x) ) dx +c)/e^(3x)`

= `(e^(6x)/6 +c)/e^(3x)`

=`(e^(6x)/6 +c)*e^(-3x)`

so `y(x)=(e^(6x)/6 +c)*e^(-3x)`

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial