What is a solution to the differential equation dy/dx=1+2xy?

Aubrie Aguilar 2022-09-22 Answered
What is a solution to the differential equation d y d x = 1 + 2 x y ?
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Answers (1)

Amiya Watkins
Answered 2022-09-23 Author has 6 answers
d y d x = 1 + 2 x y
d y d x - 2 x y = 1 ..... [1]

This is a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

d y d x + P ( x ) y = Q ( x )

This is a standard form of a Differential Equation that can be solved by using an Integrating Factor:

I = e P ( x ) d x
    = e   - 2 x   d x
    = e - x 2

And if we multiply the DE [1] by this Integrating Factor we will have a perfect product differential;

d y d x - 2 x y = 1
e - x 2 d y d x - 2 x y e - x 2 = 1 e - x 2
d d x ( y e - x 2 ) = e - x 2

This has converted our DE into a First Order separable DE which we can now just separate the variables to get;

y e - x 2 =   e - x 2   d x

The RHS integral does not have an elementary form, but we can use the definition of the Error Function :

erf ( x ) = 2 π 0 x e - t 2   d t

Which gives us:

y e - x 2 = π 2 erf ( x ) + A
y = π 2 e x 2 erf ( x ) + A e x 2
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