In this exercise you will solve the initial value problem y”...

In this exercise you will solve the initial value problem y'' - 12y' + 36y = e^{-6x}/(1 + x^2), y(0) = 6, y'(0) = 5. (1) Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y'' - 12y' + 36y = 0 is the function yh(x) = C1 y1(x) + C2 y2(x) = C1 [ ] + C2 [ ]. NOTE: The order in which you enter the answers is important; that is, C1 f(x) + C2 g(x) ? C1 g(x) + C2 f(x). (2) The particular solution yp(x) to the differential equation y'' + 12y' + 36y = e^{-6x}/(1 + x^2) is of the form yp(x) = y1(x) u1(x) + y2(x) u2(x) where u1'(x) = [ ] and u2'(x) = [ ]. (3) The most general solution to the non-homogeneous differential equation y'' - 12y' + 36y = e^{-6x}/(1 + x^2) is y = [ ] + [ ] ? [ ] dt + [ ] ? [ ] dt

Transcript

00:01 Hi, in this question we need to solve the initial value problem for y double dash minus 12y dash plus 36y is equal to it is to power minus 6x divided by 1 plus x squared and y 0 is given to be equals to 6 y dash 0 is equals to 5.

00:20 For the first part we must find the value for y 1 x and y 2 x for the solution of y h and for that we will start with writing the character 6.

00:30 Equation that is r square minus 12 r plus 36 is equal to 0 or it can be written as r minus 6 whole square is equal to 0 hence we get our roots to be equals to 6 and 6 so we can write the solution for y h x that will be equals to c1 e raise to par 6x plus c 2 x times e 6x because the roots are repeating now from here we can write our value for y 1 x that y 1 x is equal to e -r -raised -to -par 6x and y 2x is equal to x times e -raise to per 6x now as per the given equation we can write our fx to be equals to erase 2 per minus 6x divide by 1 plus x square and finding for the value of w that is the ronskian and this will be equals to y1 x that is e raised to par 6x times here we will write the derivative of this which is 6 times e raised to 6x and y2x over here x times e raised to par 6x and derivative of this over here that is 6x times e raised to par 6x plus e raised to par 6x and solving this we get this to be equals to e raised to par 12x now we can write the particular solution y p that would be equals to y1 x times ux plus which is u1 x y2x times u2x where we can find the value for u1 dash x that will be equals to function value that is fx which is negative times y2 x divide by the ronskyan and this will be further written as equals to that is fx which is errantz to par minus 6x divided by 1 plus x square it is negative times y 2 x which is x times e raise to par 6x whole divided by e raise to par 12x now solving this further we can cut this and here we were left with erase to par 6x and solving this further we will left with u1 dash x which will be equals to our final answer for u1 dash x which is negative of x times e raised to par minus 12x whole divided by 1 plus x squared.

03:05 Now similarly we can write the value for u2 dash x which will be equals to fx times y1x divide by the ronskian.

03:18 And this will be equals to fx which is e raised to par minus 6x divided by 1 plus x squared times y1x which is e raised to par 6x and it is divided by e raised to par 12x which is the ron scan value...

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