Complex roots of second-order linear ODEs with constant coefficients

[JM00404]

Problem I.
Consider the equation
$$y''+a_1y'+a_2y=0$$
where the constants $$a_1,a_2$$ are real. Suppose $$\alpha+i\beta$$ is a complex root of the
characteristic polynomial, where $$\alpha,\beta$$ are real, $$\beta\neq 0$$ .
(i) Show that $$\alpha-i\beta$$ is also a root.
(ii) Show that any solution $$\phi$$ may be written in the form
$$\phi(x)=e^{\alpha x}(d_1\cos(\beta x)+d_2\sin(\beta x))$$ ,
where $$d_1,d_2$$ are constants.
(iii) Show the $$\alpha=-a_1/2, \beta^2=a_2-(a_1^2/4)$$ .
(iv) Show that every solution tends to zero as $$x\rightarrow +\infty$$ if $$a_1>0$$ .
(v) Show that the magnitude of every non-trivial solution assumes aribtrarily large values as $$x\rightarrow +\infty$$ if $$a_1<0$$ .

Problem II.
Show that every solution of the constant coefficient equation
$$y''+a_1y'+a_2y=0$$
tends to zero as $$x\rightarrow \infty$$ if, and only if, the real parts of the roots of the characteristic polynomial are negative.

I cannot seem to solve parts (iii) through (v) in problem I and the whole of problem II correctly. I am really not all that sure where I should begin on these parts. Any assistance offered would be much appreciated. Thank you for your time.

 

[TimNguyen]

Problem I.
Consider the equation
$$y''+a_1y'+a_2y=0$$
where the constants $$a_1,a_2$$ are real. Suppose $$\alpha+i\beta$$ is a complex root of the
characteristic polynomial, where $$\alpha,\beta$$ are real, $$\beta\neq 0$$ .
(i) Show that $$\alpha-i\beta$$ is also a root.
(ii) Show that any solution $$\phi$$ may be written in the form
$$\phi(x)=e^{\alpha x}(d_1\cos(\beta x)+d_2\sin(\beta x))$$ ,
where $$d_1,d_2$$ are constants.
(iii) Show the $$\alpha=-a_1/2, \beta^2=a_2-(a_1^2/4)$$ .
(iv) Show that every solution tends to zero as $$x\rightarrow +\infty$$ if $$a_1>0$$ .
(v) Show that the magnitude of every non-trivial solution assumes aribtrarily large values as $$x\rightarrow +\infty$$ if $$a_1<0$$ .

Problem II.
Show that every solution of the constant coefficient equation
$$y''+a_1y'+a_2y=0$$
tends to zero as $$x\rightarrow \infty$$ if, and only if, the real parts of the roots of the characteristic polynomial are negative.

I cannot seem to solve parts (iii) through (v) in problem I and the whole of problem II correctly. I am really not all that sure where I should begin on these parts. Any assistance offered would be much appreciated. Thank you for your time.

For part (iii) of problem 1, have you tried to solve the auxillary equation with the arbitrary a1 and a2?

Part (iv): Look at your general solutions to the differential equation and notice what happens when you plug in "infinity" for x.

Problem 2: Solve the auxillary equation and split it into the three cases that you should know for linear, homogenous, 2nd order differential equations. Then show that only the complex root case will give you the infinity.

 

[Roti Boy]

Problem 2: Solve the auxillary equation and split it into the three cases that you should know for linear, homogenous, 2nd order differential equations.

I agree with what TimNguyen said above.
Give you another hint for question 2.
What's the value of $$|e^{i\beta x}|$$? and then what's the value of $$\lim_{x\rightarrow\infty}e^{-\lambda^2 x}$$?