hey all!
ive got two differentail equations which i really cannot do, but my friend suggested using this website so here goes!
a) xy(dy/dx) - 2(y+3)=0
b) (dy/dx) - y=xy5 (i think its a special one but i dont know how to do it!)
sorry about the writing style, i dont know how to use the special maths tool yet but i hope you understand what i mean :P
thank you
i did the manipulation slightly different from what you did, but we ended up with the same thing. i used the modified equation that Krizalid developed to plug the substitutions into. i think it works out nicer that way, but what do i know, maybe it's just nicer in my weird mind
Solve the homogeneous equation first:
What function could be? Since we know that the function must be an exponential. Thus
Plugging this into the homogeneous equation gives:
This leaves us with b = -4. So .
Now for the particular solution of . The RHS is a polynomial so we would guess that the particular solution will be a polynomial. So try . Plugging this into the differential equation gives:
So we have the simultaneous equations
and
Thus c = -1 and d = 1/4. Thus .
Since the differential equation is linear the most general solution to the differential equation will be the sum of the homogenous and particular solutions. Thus
Now, to get y(x) we know that . Thus we finally have that
-Dan