Problem 1: Solve the following ordinary differential...

Problem 1: Solve the following ordinary differential equation : 7 d^2y/dx^2 - 2 dy/dx - y + x = 0 with the boundary conditions y(0) = 5 and y(20) = 8, using the finite-difference approach with h = 2. a) Draw the discretized domain with the nodes. Using centered differences for the first and second derivatives, write an approximate expression for the ODE at the interior nodes xi. This generic expression should be valid for all interior nodes. b) Write the system of equations (i.e. each equation at each interior node). c) Write the system in matrix format and solve in Matlab using the left matrix divide command d) Plot your results in Matlab.

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0:00 Hello students.

00:01 Today we will discuss about this question.

00:04 In this question we are given that we need to solve the following ordinary differential equation, that is 7d2 d squared by dx square y divided by d x minus 2 divide divided by d x minus y plus x is equal to 0 with a boundary condition that is y of 0 is equal to 5, y of 20 is equal to 8, using a finite difference approach with h is to 2.

00:31 And here we need to draw the discretized domain with the nodes using the centered difference for the first and second derivative and write the approximate expression for the ode at the interior nodes xi and generic expression should be valid for the all interior nodes.

00:55 So here we need to write the system of equation.

01:00 That is equal to question mark.

01:07 So here, first of all, we can write that y -double -y -dash -y -dash -of -x -i that is equal to y -of -x -i plus 1 minus y -of -x -i -minus 1 divided by 2 -h, y -double -dash -of -xi that is equals to y -of -x1 plus y -dash plus minus 2y x -1 plus y -1 plus y of, x i minus 1 divided by h so therefore here x i that is equal to o plus p h therefore x i that is equal to i h that is equal to i 2 so here t of y x i plus 1 minus 2 y x i plus y x i minus 2 y x i plus y x i minus 1 divided by h2 minus 2 y x i plus 1 minus y x i minus 1 divide by 2 h minus y x i minus 1 minus y x i plus x i that is equals to 0 so therefore h that is equal to 2 now 7 divided by 4 y x i plus 1 minus 2 y x i plus y x i minus 1 minus 2 divided by 4 y x i plus 1 minus y x i minus 1 minus y x i minus 1 minus y x i plus i 2 that is equals to 0 so here we can write 9 y x i minus 1 minus 18 y x i plus 5 y x i plus 8 i is equal to 0...

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