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March 4th, 2019, 12:14 AM  #1 
Newbie Joined: Mar 2019 From: Holland Posts: 1 Thanks: 0  Help with differential equation
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 $\displaystyle \frac{d^2F}{dX^2}=\frac{2\ast\sigma_s}{E\ast\left[\sqrt{4.65152d^226.853\frac{M}{\sigma_sd}0.41955d}\right]}$ 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: $\displaystyle \frac{d^2F}{dx^2}=\frac{c}{e\sqrt{abX}}$ Boundary conditions: $\displaystyle x=\frac{L}{2}$, $\displaystyle \frac{dF}{dX}=0$ Solving this: $\displaystyle dF/dx=\frac{2c\ast\left(e\ast\ln{\left(\sqrt{abx}\ast e\right)}+\sqrt{abx}\right)}{b}+C_1$ $\displaystyle C_1=\frac{2c\ast\left(e\ast\ln{\left(\sqrt{ab\frac{L}{2}\ }\ast e\right)}+\sqrt{ab\frac{L}{2}}\right)}{b}$ 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. $\displaystyle F=2\frac{c}{b}\left\{\frac{2}{3}\frac{\left(abX\right)^\frac{3}{2}}{b}\frac{e}{b}\left[ln\left(\sqrt{abX}e\right)\left(abXe^2\right)+\frac{3}{2}e^2e\sqrt{abX}\frac{abX}{2}\right]\ \right\}+C_1X+C_2$ $\displaystyle C_1=\frac{2c}{b}\left[\sqrt{ab\frac{L}{2}}+e*ln\left(\sqrt{ab\frac{L}{2}}e\right)\right]$ $\displaystyle C_2=Y_0\frac{2c}{b}\left\{\frac{2}{3}\frac{\left(abX\right)^\frac{3}{2}}{b}\frac{e}{b}\left[\ln{\left(\sqrt{abX_0}e\right)\left(abX_0e^2\right)+\frac{3}{2}e^2e\sqrt{abX_0}\frac{abX_0}{2}}\right]\right\}K_1X_0$ Many thanks in advance. Last edited by skipjack; March 4th, 2019 at 03:43 AM. 

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