difficult to solve some questions : Linear Algebra and Differential Equations

[mallange87]

1. Solve y''(x) + 9y(x) = 0 where y(0) + y'(0) = 0 and y(π) − y'(π) = 10. Plot the solution on the domain0 < x < π


2. Given that y1(x) and y2(x) satisfy the differential equation f₁(x)y''(x) + f₂(x)y'(x) + f₃(x)y(x) = 0 show that y(x) = C₁y₁(x) + C₂y₂(x)also satisfies this differential equation.


3. Find the Laplace transform of tⁿf'(t) where n ≥ 1


4. The fundamental set solutions of a third order ODE whose characteristic polynomial has one real root and two complex roots is
y₁(t) = e ^λ₁t, y₂(t) = e^αtcos(βt), and y₃(t) = e^αtsin(βt)
where λ_1,2 = α ± βi. Prove that y₁(t), y₂(t), and y₃(t) indeed form the fundamental set of solutions, i.e. prove they are linearly independent.

 

[Deleted member 4993]

1. Solve y''(x) + 9y(x) = 0 where y(0) + y'(0) = 0 and y(π) − y'(π) = 10. Plot the solution on the domain0 < x < π


2. Given that y1(x) and y2(x) satisfy the differential equation f₁(x)y''(x) + f₂(x)y'(x) + f₃(x)y(x) = 0 show that y(x) = C₁y₁(x) + C₂y₂(x)also satisfies this differential equation.


3. Find the Laplace transform of tⁿf'(t) where n ≥ 1


4. The fundamental set solutions of a third order ODE whose characteristic polynomial has one real root and two complex roots is
y₁(t) = e ^λ₁t, y₂(t) = e^αtcos(βt), and y₃(t) = e^αtsin(βt)
where λ_1,2 = α ± βi. Prove that y₁(t), y₂(t), and y₃(t) indeed form the fundamental set of solutions, i.e. prove they are linearly independent.

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

 

[HallsofIvy]

I have always found that it is very difficult, indeed impossible, to solve any problem if you don't try!

1. Solve y''(x) + 9y(x) = 0 where y(0) + y'(0) = 0 and y(π) − y'(π) = 10. Plot the solution on the domain0 < x < π

This problem is a homogenous linear differential equation with constant coefficients. Have you not learned how to solve those? What is its "characteristic equation"?

2. Given that y1(x) and y2(x) satisfy the differential equation f₁(x)y''(x) + f₂(x)y'(x) + f₃(x)y(x) = 0 show that y(x) = C₁y₁(x) + C₂y₂(x)also satisfies this differential equation.

To show that a given function satisfies a given differential equation, find its derivatives and put them into the equation. I presume you know that the first and second derivatives of \(\displaystyle y= C_1y_1+ C_2y_2\) are \(\displaystyle y'= C_1y_1'+ C_2y_2'\) and \(\displaystyle y''= C_1y_1''+ C_2y_2''\). What do you get when you put those into the differential equation?

3. Find the Laplace transform of tⁿf'(t) where n ≥ 1

Do you not know what the "Laplace transform" is? The Laplace transform of f(x) is \(\displaystyle \int_0^\infty f(t)e^{-st}dt\) so you need to integrate \(\displaystyle \int_0^\infty t^n f'(t)e^{-st}dt\). You should be able to integrate "by parts". To start, I would let \(\displaystyle u= t^n e^{-st}\) and \(\displaystyle dv= f'(t)\)

4. The fundamental set solutions of a third order ODE whose characteristic polynomial has one real root and two complex roots is
y₁(t) = e ^λ₁t, y₂(t) = e^αtcos(βt), and y₃(t) = e^αtsin(βt)
where λ_1,2 = α ± βi. Prove that y₁(t), y₂(t), and y₃(t) indeed form the fundamental set of solutions, i.e. prove they are linearly independent.

Do you know what "linearly independent" means? Three functions, f, g, and h, are "linearly independent" if and only if af(x)+ bg(x)+ ch(x)= 0, for all x, implies a= b= c= 0.

In any case, why are you asking people to do your home work for you? You should, at the very least, show what you have attempted and where, specifically, you are having difficulty.