What is the general solution of the differential equation dy/dx−2xy=x?

Lindsey Ibarra 2022-09-27 Answered
What is the general solution of the differential equation d y d x - 2 x y = x ?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

brodireo1
Answered 2022-09-28 Author has 12 answers
d y d x - 2 x y = x

Which we can write as:

d y d x = 2 x y + x
d y d x = ( 2 y + 1 ) x
1 2 y + 1 d y d x = x

Which is a first order separable differential equation, so we can "separate the variables" to get:

  1 2 y + 1   d y =   x   d x

Integrating we get, the General Solution:

1 2 ln | 2 y + 1 | = 1 2 x 2 + C

Applying the initial condition y(1)=1 we find:

1 2 ln 3 = 1 2 + C C = 1 2 ln 3 - 1 2

So we can write an implicit particular solution as:

1 2 ln | 2 y + 1 | = 1 2 x 2 + 1 2 ln 3 - 1 2

We typically require an explicit solution, so we can rearrange as follows:

ln | 2 y + 1 | = x 2 + ln 3 - 1
| 2 y + 1 | = e x 2 + ln 3 - 1

Noting that the exponential function is positive over its entire domain, (as e x > 0 x ):

2 y + 1 = e x 2 - 1 e ln 3

2 y = 3 e x 2 - 1 - 1

y = 3 2 e x 2 - 1 - 1 2
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-06-18
I have the following differential equation problem but I couldn't proceed any further -
d y d x = a   y x 1   b   ( 1 x ) n ( y 1 y ) m
where, x [ 0 , 1 ]   and   y [ 0 , 1 ]
But I can't solve it down.
I have tried y = u y 1       where ,   y 1 = x n m  , but it didn't help.
Wolfram gives the solution as -
y ( x ) = c 1 exp ( a x b x m n + 1 ( x + 1 ) m d x )
How to simplify the integral?
I just wanted a hint that whether it can be solved? If yes, please just tell me what am I doing wrong.
asked 2022-09-29
What is a particular solution to the differential equation d y d x = e x - y with y(0)=2?
asked 2022-09-29
How do you solve for xy'−y=3xy given y(1)=0?
asked 2022-04-10
For every differentiable function f : R R , there is a function g : R × R R such that g ( f ( x ) , f ( x ) ) = 0 for every x and for every differentiable function h : R R holds that
being true that for every x R , g ( h ( x ) , h ( x ) ) = 0 and h ( 0 ) = f ( 0 ) implies that h ( x ) = f ( x ) for every x R .
i.e every differentiable function f is a solution to some first order differential equation that has translation symmetry.
asked 2022-06-01
How can I start to solve this differential equation?
y = y 2 y ln ( y ) + y x
asked 2020-11-26

Solve differential equation (x2lnx)y+xy=0

asked 2021-03-07
Solve differential equationxu(x)=u24