How to I solve homogeneous equation of #y dy/dx +x=2y# ?
3 Answers
The general solution is
Explanation:
The ODE is
Divide by
Let
Then,
Substituting in the ODE
Therefore,
Integrating both sides
For the
Let
The general solution is
Where
Substitute
Explanation:
Be careful with terminology here - "homogeneous ordinary differential equation" can mean two entirely different things!
1) An equation in
2) A first-order ODE where
In this question we are dealing with the second meaning; let's rearrange into the needed form:
Substitute
so
Substituting in:
This is now a separable equation, so integrate for the solution:
The
Put the two integral solutions together:
Now substitute back for
Some tidying up rearrangement:
As we have both
The general solution of given equation is :
Explanation:
Here,
Subst.
So,
Integrating both sides:
Subst. back ,
This is the general solution of given equation.