# 4th Order Linear ODE

#### alane1994

##### Active member
Find the solution of the given initial value problem, and plot its graph. How does the solution behave as $$t\rightarrow\infty$$

$$y^{(4)}-4y'''+4y''=0$$

My work, which coincidentally I believe is incorrect....

From the above differential equation,

$$r^4-4r^3+4r^2=0$$

$$r^2(r-2)^2=0$$

$$r=0,~2$$

Then, would I just input that into a form like this?

$$y=c_1+c_2e^{2t}+c_3te^{2t}$$

I definitely feel as though this isn't correct...

#### chisigma

##### Well-known member
Find the solution of the given initial value problem, and plot its graph. How does the solution behave as $$t\rightarrow\infty$$

$$y^{(4)}-4y'''+4y''=0$$

My work, which coincidentally I believe is incorrect....

From the above differential equation,

$$r^4-4r^3+4r^2=0$$

$$r^2(r-2)^2=0$$

$$r=0,~2$$

Then, would I just input that into a form like this?

$$y=c_1+c_2e^{2t}+c_3te^{2t}$$

I definitely feel as though this isn't correct...
Setting $\displaystyle y^{\ ''}= z$ the ODE becomes...

$\displaystyle z^{\ ''} - 4\ z^{\ '} + 4\ z\ (1)$

... the solution of which is...

$\displaystyle z(t)= c_{1}\ e^{2\ t} + c_{2}\ t\ e^{2\ t}\ (20)$

Now You can solve the ODE...

$\displaystyle y^{ ''} = z\ (3)$

... with two successive integration obtaining...

$\displaystyle y^{\ '} = \int z(t)\ dt\ (4)$

$\displaystyle y = \int y^{ '} (t)\ dt\ (5)$

Kind regards

$\chi$ $\sigma$

#### alane1994

##### Active member
Thank you very much! That makes quite a bit more sense!

#### MarkFL

Staff member
Find the solution of the given initial value problem, and plot its graph. How does the solution behave as $$t\rightarrow\infty$$

$$y^{(4)}-4y'''+4y''=0$$

My work, which coincidentally I believe is incorrect....

From the above differential equation,

$$r^4-4r^3+4r^2=0$$

$$r^2(r-2)^2=0$$

$$r=0,~2$$

Then, would I just input that into a form like this?

$$y=c_1+c_2e^{2t}+c_3te^{2t}$$

I definitely feel as though this isn't correct...
Both of your characteristic roots are repeated, so the general form of your solution would be:

$$\displaystyle y(t)=c_1+c_2t+c_3e^{2t}+c_4te^{2t}$$