Solution of set of non linear coupled 4th order ode with algebraic equation

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Hello Everybody and Good Morning.

I have set of three coupled equations that need to be solved simultaneously through matlab code to obtain results for six variables. It is actually system of equations with one 4th order equation and one algebraic equation.The details about the problem description can be found as: enter image description here

The expected Results of this program should be as following: enter image description here

Please have a kind look to my problem and help me to figure out with the possible alogrithm of the matlab code . If you can provide me with example code, i will be more than thankful and indebted to you.

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