Differential equation of second order with two variables
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to be solved is the equation 𝐸𝐼𝑤4(𝑥,𝑡)+𝑚 𝑤̈(𝑥,𝑡)=0 and this can be solved by expressing 𝑤(𝑥,𝑡)= 𝜙(𝑥)⋅𝑌(𝑡)
for 𝜙(𝑥) we do have this expression 𝜙𝑖(𝑥)=𝐴⋅[sin(𝜆𝑖 𝑥𝐿)−sinh(𝜆𝑖 𝑥𝐿)+sin(𝜆𝑖)+sinh(𝜆𝑖)cos(𝜆𝑖)+cosh(𝜆𝑖)⋅(cosh(𝜆𝑖 𝑥𝐿)−cos(𝜆𝑖𝑥𝐿)) ]
and for 𝑌𝑖(𝑡)=𝑌(0)cos(𝜔𝑖 𝑡)+𝑌̇(0)𝜔sin(𝜔𝑖(𝑡))=𝐶(0)⋅𝑐𝑜𝑠(𝜔𝑖 𝑡+𝜃(0)). Whereas 𝜆𝑖 is solved through this equation 1+cos(𝜆𝑖)⋅cosh(𝜆𝑖)=0
I am guessing that to solve the first differential equation as conditions we can use the three other equations given.
Does anyone has some tipps how this differential equation can be solved?
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Pranjal Saxena
am 28 Jul. 2023
Hi Florian,
I understand that you want to solve this second order differential equation.
You can use the “Symbolic Math Toolbox” in MATLAB to do so.
A typical approach to solving higher-order ordinary differential equations is to convert them to systems of first-order differential equations, and then solve those systems.
First you need to convert the second-order differential equation into a system of differential equations that can be solved using the numerical solver “ode45” of MATLAB.
You can refer to the following MATLAB documentations for more information:
I hope this helps.
Warm Regards,
Pranjal.
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