Differential Equations Ordinary and Partial Differential Equations Math Forum

March 4th, 2019, 01:14 AM   #1
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Hi, I'm new to this forum. Found this forum searching for help with a Math question. I need some help with a differential equation that I can't solve. I hope you guys can help me.

The equation
$\displaystyle \frac{d^2F}{dX^2}=-\frac{2\ast\sigma_s}{E\ast\left[\sqrt{4.65152d^2-26.853\frac{M}{\sigma_sd}-0.41955d}\right]}$

With the following replacements

$\displaystyle M=\frac{P}{2}\ast x$

$\displaystyle a=4.65152d^2$

$\displaystyle b=\frac{26.853}{2}\frac{P}{\sigma_s\ast d}$

$\displaystyle c=2\frac{\sigma_s}{E}$

$\displaystyle e=0.41955d$

We can rewrite the equation as: $\displaystyle \frac{d^2F}{dx^2}=\frac{c}{e-\sqrt{a-bX}}$

Boundary conditions: $\displaystyle x=\frac{L}{2}$, $\displaystyle \frac{dF}{dX}=0$

Solving this:

$\displaystyle dF/dx=\frac{2c\ast\left(e\ast\ln{\left(\sqrt{a-bx}\ast e\right)}+\sqrt{a-bx}\right)}{b}+C_1$

$\displaystyle C_1=-\frac{2c\ast\left(e\ast\ln{\left(\sqrt{a-b\frac{L}{2}\ }\ast e\right)}+\sqrt{a-b\frac{L}{2}}\right)}{b}$

Next we say the boundary conditions are:

$\displaystyle x=x_0$

$\displaystyle F=F_0=\frac{P}{d^4\ast E}\ast(\frac{4}{\pi}L^2\ast\ x_0-\frac{16}{3\pi}\ast\ x_0^3)$

with $\displaystyle x_0=\pi/16\ast\sigma_s/P\ast d^3$

This is the point where I don't know how to solve it anymore. I know what the solution is, but I want to know how they got there.

$\displaystyle F=2\frac{c}{b}\left\{-\frac{2}{3}\frac{\left(a-bX\right)^\frac{3}{2}}{b}-\frac{e}{b}\left[ln\left(\sqrt{a-bX}-e\right)\left(a-bX-e^2\right)+\frac{3}{2}e^2-e\sqrt{a-bX}-\frac{a-bX}{2}\right]\ \right\}+C_1X+C_2$

$\displaystyle C_1=-\frac{2c}{b}\left[\sqrt{a-b\frac{L}{2}}+e*ln\left(\sqrt{a-b\frac{L}{2}}-e\right)\right]$

$\displaystyle C_2=Y_0-\frac{2c}{b}\left\{-\frac{2}{3}\frac{\left(a-bX\right)^\frac{3}{2}}{b}-\frac{e}{b}\left[\ln{\left(\sqrt{a-bX_0}-e\right)\left(a-bX_0-e^2\right)+\frac{3}{2}e^2-e\sqrt{a-bX_0}-\frac{a-bX_0}{2}}\right]\right\}-K_1X_0$

Attached Images equation.JPG (94.4 KB, 1 views)

Last edited by skipjack; March 4th, 2019 at 04:43 AM. Tags differential, equation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post max233 Calculus 4 March 26th, 2016 04:21 AM Sonprelis Calculus 6 August 6th, 2014 11:07 AM consigliere Differential Equations 5 August 23rd, 2013 11:54 AM PhizKid Differential Equations 0 February 24th, 2013 11:30 AM JohnC Differential Equations 4 March 9th, 2012 12:02 AM

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