MHB Solve 4th Order Linear ODE and Plot Graph: t→∞

[alane1994]

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]

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]

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

 

[MarkFL]

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:

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