# Urgent help Homogeneous Linear D.E.

## Homework Statement

A linear equation in form:

dy/dx + P(x)y = 0 is said to be homogeneous since Q(x)=0.

a) show that y=0 is a trivial solution (wasn't even taught what a trivial solution is)
b) show that y=y1(x) is a solution and k is a constant, then y=ky1x is also a solution.
c) show that if y=y1x and y=y2x are solutions, then y=y1x + y2x is a solution

I don't even know how to start this problem. For part a) i simply plugged in 0 for y and got dy/dx=0 . doesn't seem right. then i tried separation of variable and got stuck at

(1/y)dy=-P(x)dx

can someone please guide me through? i have about three other problems like this and i haven't got a clue how to solve them.

p.s. that's y(sub1) and y(sub2)

you're not solving anything specific, since you don't really know about your functions. you're just proving general cases.

the trivial solution is just y=0. that's certainly true, dy/dx of 0 = 0, and P(x)y=P(x)0=0.

so, if you know that y1 is a solution, what do we know about constants? we can pull them out and apply them after we differentiate. so that's just like multiplying each term by k, and k0 = 0.

so we know that y1 and y2 are solutions in themselves. if you plug in y1 + y2 as a solution, well,
dy/dx (y1 + y2) = dy/dx y1 + dy/dx y2.
and P(x)(y1 + y2) = P(x)y1 + P(x) y2.
so dy/dx (y1 + y2) + P(x)(y1 + y2) = dy/dx y1 + dy/dx y2 + P(x)y1 + P(x) y2 = dy/dx y1 + P(x)y1 + dy/dx y2 + P(x)y2.
we know that dy/dx y1 + P(x)y1 = 0, and we know that dy/dx y2 + P(x)y2 = 0, since they're both solutions, so then dy/dx y1 + P(x)y1 + dy/dx y2 + P(x)y2 = 0.
i hope that's right, i'm a bit rusty with this stuff.