Q$1$ By substituting $(2)$ into the ODE, verify that the general solution to equation $(1)$ is givenby $=$ c$1$ cos$($t$)+$ c$2$ sin$($t$)(2)$where $=$pg$/`.$The double pendulumA double pendulum consists of two pendulums attached end$-$to$-$end, as shown below$\mathrm{.}12``$BA$2$This system exhibits rich dynamic behaviour, which in some situations is chaotic. For thisproject, we will consider the case where the two bobs have equal mass, the two rods are ofequal length, $`,$ and the oscillations are small. For small oscillations $($that is$,1$ and $2$ small$),$the behaviour is not chaotic and the motion of the double pendulum is governed by the followingset of coupled ODEs:$1=221+222=221222(3)$where $=$pg$/`$ and $/61/6,/62/6.$ For this project, you should assumeg $=9\mathrm{.}8$ ms$2$ and the length of each rod, $`=10$ m$.$Q$2$: Write this system of equations in matrix form: $00=2$A$,$ where $="12$#$.$Since the pendulum will oscillate back and forth, we should expect the solutions to this systemof equations to contain oscillatory functions.Q$3$: Both $=$ x cos$($ t$)$ and $=$ x sin$($ t$),$ where x $="$x$1$x$2$#$,$ are solutions to thesystem of equations given by $(3).$ By substituting each solution into the matrix equation fromQ$2,$ state what relationship x and have to A$.$Q$4$: In your Matlab script file, write the code required to find the eigenvalues $(1$ and $2)$ andthe corresponding eigenvectors $($x$1$ and x$2)$ for the matrix A$.$ Use the corresponding sectionof the skeleton Matlab script at the end of this assignment. Run your script file to find$1,2,$ x$1$ and x$2.$ Write your answers in your report.The general solution of the system of equations $(3)$ will be a linear combination of all of thesolutions described above.Q$5$: Write the general solution in terms of the eigenvalues and eigenvectors of the matrix A$.$Hint: The system of equations $(3)$ consists of two second order linear ODEs. This means thatwe should expect the general solution to have four arbitrary constants, that is$,$ the general solution should be a linear combination of four solutions. Use c$1,$ c$2,$ c$3,$ c$4$ for the arbitrary constants.The values of c$1,$ c$2,$ c$3,$ c$4$ will depend on the initial conditions of the system, that is$,$ the specificinitial value problem. We will consider the following specific initial value problem:$1(0)=/12$ rad $2(0)=0$ rad$1(0)=/6$ rad s$12(0)=/12$ rad s$1(4)$Q$6$: Give a physical interpretation of the initial conditions listed above in $(4)\mathrm{.}3$Q$7$: To determine the unknown constants for any initial value problem, we can write a systemof linear equations $($for unknowns c$1,$ c$2,$ c$3,$ c$4)$ and solve a matrix equation of the form Bc $=$ dwhere c $=[$c$1,$ c$2,$ c$3,$ c$4]$T$.$ Write down B and d in your report. Then, in your Matlab scriptfile, write the code required to find c$1,$ c$2,$ c$3,$ c$4$ for the specific initial value problem describedabove. Be sure, though, to set it up so that you can easily change the initial value problem.Our solution, $,$ describes the angles between the vertical and each rod at a particular time t$.$However, to construct our animation, we must first determine the positions of the bobs at timet. Denote the position of bob A by $($ax$,$ ay$)$ and the position of bob B by $($bx$,$ by$).$Q$8$: Fill in the necessary Matlab code required to calculate the positions of the bobs at times$0$ t $20$ seconds $($use a stepsize of $0\mathrm{.}05$s$).$ Provide supporting handwritten workings thatshow how a$\_$x$,$ a$\_$y$,$ b$\_$x$,$ b$\_$y are determined.To create the animation, we use the for loop within the skeleton Matlab code provided. Copythis to your Matlab script file.Q$8$: Run the animation. Explain precisely what the lines of code marked with $(***)$ do$.$ Printout the position of the double pendulum at t $=20$s and include this figure in your report $($Figure$1).$Q$9$: Create a second Matlab figure that shows the path traced out by each bob. Show bob Ain red and bob B in blue. Include a printout of this figure in your report $($Figure $2).$ Explainwhy this does not need to be done inside the animation loop.Q$10$: Create a third Matlab figure that plots $2$ against $1.$ Be sure to label your axes andinclude a printout of this figure in your report $($Figure $3).$ This is an example of a Lissajouscurve. $($You do not need to know this, but you might be interested in the graph that it creates!$)$Q$11$: Choose your own initial value problem $($remember to choose appropriate values for $1$ and$2)$ and create the appropriate $3$ Matlab figures by running your script file.