Is the Sum of Solutions of Linear Differential Equations Also a Solution?

[Naeem]

Q. If y = y1 (x) is a solution of dy/dx + p(x) y = r(x) and y = y2(x) is a solution of dy/dx + p(x) y = q(x), then y = y1(x) + y2(x) is a solution of

dy/dx + p(x) y = q (x) + r(x).

I know that the a Linear differential equation is of the form,

dy/dx + p(x) y = q(x)

Any thoughts on how to proceed with this one.

Would appreciate, ideas.

 

[Hurkyl]

Please read the guidelines for posting homework help questions.

https://www.physicsforums.com/showthread.php?t=4825

You must have had a thought on something to do, even if it is something as trivial, such as rewriting a statement in terms of a definition.

 

[HallsofIvy]

Q. If y = y1 (x) is a solution of dy/dx + p(x) y = r(x) and y = y2(x) is a solution of dy/dx + p(x) y = q(x), then y = y1(x) + y2(x) is a solution of

dy/dx + p(x) y = q (x) + r(x).

I know that the a Linear differential equation is of the form,

dy/dx + p(x) y = q(x)

Any thoughts on how to proceed with this one.

Would appreciate, ideas.

HOW do you show that a given function IS a solution to a differential equation? Show us what you have done or what you DO understand about this.

 

[dvs77]

d(y1+y2)/dx+P(y1+y2)=(dy1/dx+py1)+(dy2/dx+py2)
=q(x)+r(x)
since q and r are solutions of those non homogeneous differential equations