linear Differential equations solving

[Raja88]

Hi everyone, I am new to differential equations

I have two second order differential equations with boundary conditions as shown in the attached figure. The author has generated a graph between Ɵ and t as a result of solving these equations. I need clarifications on how the author got the graph with the two differential equations. Thanks in advance

 

[Raja88]

Solving second order differential equations

Hi everyone,

I have two second order differerential equations as given in the attached figure with its boundary conditions. The authors has the result in form of a graph between theta and t. I need clarification in solving the equations and plotting the graphs . Thanks in advance

 

[HallsofIvy]

The first thing you should see is that if you subtract one equation from the other, the two "\(\displaystyle \xi''\)" terms cancel leaving a second order linear differential equation with constant coefficientsd equal to a non-homogeneous part that involves \(\displaystyle \xi\). You can solve that for \(\displaystyle \theta\), using "variation of parameters", as a function of t and \(\displaystyle \xi\). Then put that into either equation to get a linear differential equation for \(\displaystyle \xi\) with constant coefficients