What is the general solution of the differential equation dy/dx−2y+a=0?

tamolam8 2022-09-12 Answered
What is the general solution of the differential equation d y d x - 2 y + a = 0 ?
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Answers (1)

Raphael Singleton
Answered 2022-09-13 Author has 19 answers
First write the DE in standard form:

d y d x - 2 y + a = 0
d y d x - 2 y = - a ......[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   d x
    = e - 2 x

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

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

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

y e - 2 x =   - a e - 2 x   d x

Which we can easily integrate to get:

y e - 2 x = 1 2 a e - 2 x + C
y = 1 2 a + C e 2 x
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