Differential equation of second order with two variables

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?

Answer 1

[Pranjal Saxena]

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:

Solve Differential Algebraic Equations (DAEs) - MATLAB & Simulink (mathworks.com)

Solve Differential Equation - MATLAB & Simulink (mathworks.com)

Solve a System of Differential Equations - MATLAB & Simulink (mathworks.com)

Solve a Second-Order Differential Equation Numerically - MATLAB & Simulink (mathworks.com)

I hope this helps.

Warm Regards,

Pranjal.