# Dif. eq. problem - dont know why it is wrong

dif. eq. problem -- dont know why it is wrong

I did this problem three times with the same wrong answer...

solve for y:
$$y'''-9y''+18y'=0$$
y(0)=5
y'(0)=7
y''(0)=6

so I change it into the auxiliary equation then solve for the variables, i get the roots: 3 and 6. so the general solution would be:

$$y'=c_1e^{3x}+c_2e^{6x}$$
solving for c1 and c2 i get:

$$7=c_1+c_2$$
$$2=c_1+2c_2$$
$$c_1=12$$
$$c_2=-5$$

$$y=\int{12e^{3x}-5e^{6x}}$$
$$y=36e^{3x}-30e^{6x}+c$$

solving for c when x=0 and y=5, i get

$$y=36e^{3x}-30e^{6x}-1$$

what am i donig wrong?

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
Hey! y'''- 9y"+ 18y'= 0 is a third order differential equation!
Its auxiliary equation, r3- 9r2+ 18r= r(r2- 9r+ 18)= r(r- 3)(r- 6) has three roots! They are 0, 3, and 6.

The general solution for y is y(x)= C1e3x+ C2e6x+ C3 (since e0x= e0= 1). Use that to satisfy the initial conditions:
y(0)= C1+ C2+ C3= 5. y'(0)= 3C1+ 6C2= 7, and y"(0)= 9C1+ 36C2= 6.

That's how I would have done it. What you did was treat it as a second order d.e. for y' and then integrate. You would have gotten exactly right answer except the anti-derivative of ea is (1/a)eax, not aeax!

jamesrc
Science Advisor
Gold Member
Double check your integration:

$$\int 12 e^{3x}=12\frac{e^{3x}}{3}+c$$

and so on...