Question

Proper way to rearrange this ODE.

Answer 1

\[A(B)\frac{ d^{2}y }{ dx ^{2} } = -P(L-x)\]

Answer 2

We want \[\frac{ dy }{ dx }\] on one side of the equation.

Answer 3

@ganeshie8 Could you help here? I need to solve this so I can implement Euler's Method.

Answer 4

whats A(B) ?

Answer 5

Two variables that are being multiplied together.

Answer 6

send them to the other side by dividing AB ?

Answer 7

then integrate maybe ?

Answer 8

Okay. I'll give it a try.

Answer 9

If \(A,B,P,L\) are all constants, you can find the characteristic equation, solve for its roots, and determine the general solutions.

Alternatively, you can reduce the order to get a system of first-order derivatives.

Answer 10

How do I reduce the order?

Answer 11

Do you know how to solve a system of linear ODEs? There's not much point in the order-reduction if it involves techniques you're not familiar with.

Answer 12

You wanted to solve it numerically right ?

Answer 13

Oh didn't notice the Euler method requirement. Sorry. There's a form of the method that uses \(y''\).

Answer 14

I'll post the full question.A 4.0 m long uniform cantilever beam has a 20 kN point load (P) applied at its free end. The basic
ODE for the elastic curve for the beam is: (see equation)

where the modulus of elasticity E = 2.00 x 1011 N/m2
, the area moment of inertia Izz = 7.00 x 10-5 m4
, and
L = 4.0 m.

Using Euler’s method solve for the deflection (y) along the beam from x= 0 to x= 4.0 m. Use a step
size of 0.25 m. The appropriate initial conditions at x= 0 are y= 0, and dy/dx= 0. Include hand calculations
for at least the first two steps of the solution.

Answer 15

elasticity should be A = 2.00 * 10^11
area moment of inertia should be B = 7.00 * 10^-5

Answer 16

I should correct myself: reduction of order could indeed be very useful here. Try setting \(y'=v\), so that \(y''=v'\).

This means \(v(x_0)=y'(x_0)\), which is presumably equal to 0 judging by the initial conditions you listed.

Answer 17

Okay, I'm gonna look through my text book and online to see how to reduce the order. I am not very familiar with it.

Answer 18

Here's a good resource

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/ode/second/so_num/so_num.html

Answer 19

Awesome. Thank you, I should be fine now.

Answer 20

Really need help. Thought I could do it, but I'm really stuck.

Answer 21

@Directrix @Luigi0210 @.Sam. @dan815 @Compassionate @mathslover Could anyone help here?

Answer 22

@Abhisar Help?

Answer 23

\[ABy''(x)=-P(L-x)\]
Set \(v=y'\), then
\[ABv'(x)=-P(L-x)~~\iff~~v'(x)=\frac{P(x-L)}{AB}\]
Now you have two ODEs to consider, both of which can be solved numerically with Euler's method:
\[\begin{cases}
y'=v&\text{with }y(0)=0\\\\
v'=\dfrac{P(x-L)}{AB}&\text{with }v(0)=y'(0)=0
\end{cases}\]
You need to solve for \(v\) before you can solve for \(y\).

Answer 24

Formula for Euler's method: If \(y'=v\) and \(v'=f(x,y,v)\) with step size \(h\),
\[\begin{cases}y_{n+1}=y_n+hv_n\\
v_{n+1}=v_n+hf(x_n,y_n,v_n)\\
x_{n+1}=x_n+(n+1)h\\
x_0=y_0=v_0=0\end{cases}\]
Here's a table to get you started.
\[\begin{array}{c|c|c|c}
n&x_n&v_n&y_n&f(x_n,y_n,v_n)\\
\hline
0&0&0&0&-\frac{PL}{AB}\\\\
1&0.25&-\frac{PL}{4AB}&-\frac{PL}{16AB}&\frac{P(1-4L)}{4AB}\\\\
2&0.5&-\frac{PL}{4AB}+\frac{P(1-4L)}{16AB}&-\frac{PL}{16AB}-\frac{PL}{16AB}&\frac{P(1-2L)}{2AB}\\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\\
16&4.0&\bullet&\bullet&\bullet
\end{array}\]