For second order DEs, the roots of the characteristic...

For second order DEs, the roots of the characteristic equation may be real or complex. If the roots are real, the complementary solution is the weighted sum of real exponentials. Use C1 and C2 for the weights, where C1 is associated with the root with smaller magnitude. If the roots are complex, the complementary solution is the weighted sum of complex conjugate exponentials, which can be written as a constant times a decaying exponential times a cosine with phase. Use this representation when dealing with complex roots, in which case C1 is the constant and Phi is the phase. (Note: Some equations in some textbooks give the constant multiplying the decaying exponential as 2C1. The constant for this problem should be C1 alone.) All numerical angles / phases should be given in radian (not degrees). Do not forget to include the step function u(t) as part of your answer. Given the differential equation y'' + 2y' + 10y = 2u(t) . a. Write the functional form of the complementary solution, yc(t). yc(t) = C1*e^(-t)*cos(3*t+Phi)*u(t) help (formulas) b. Find the particular solution, yp(t). yp(t) = 1/5 help (formulas) c. Find the total solution, y(t) for the initial conditions y(0) = 4 and y'(0) = 7. y(t) = (19/5)*e^(-t)*cos(3*t+Phi)*u(t) + (1/5) help (formulas)

Transcript

00:01 In this question we are given a differential equation that is y double dash plus 2y dash plus 10y is equal to 2 u t and in the a part we have to find the complex we have to find the complementary solution.

00:18 So in the a part that is represented by y c t so basically this equation is d2 y over d t 2 plus 2 d y over d t plus 10y is equal to 2 u t now let d over d t as capital d so we have d square y plus 2 d y plus 10 y is equal to 2 u t so we have d square plus 2d plus 10 times y is equal to 2 u t auxiliary equation is auxiliary equation is d square plus 2d or we can say here auxiliary equation is here m square plus 2m plus 10 equal to 0 so here we have m as minus 2 plus minus root of 2 squared.

01:34 There is 4, minus 4 multiplied by 1 multiplied by 10 over 2 multiplied by 1, that is 2.

01:40 So m is equal to minus 2 plus minus root of minus 36 over 2.

01:49 So m is equal 2 plus minus.

01:54 Here it is 6 .i over 2 so m is equal to minus 1 plus minus 7.

02:03 So m is equal to 3i now since the root are complex in nature so the root of the ogret equation complex so the complementary function is weighted sum of complex conjugate exponential which can be written as a constant time a decaying exponential so y c t is decaying time exponential times a cosine function with phase so y c t equal to c1 e to the bar minus t cosque 3 t plus 5 multiplied by u t this is our complementary function now in the b part we have to find the particular solution and here c1 the value c1 and 5 can be found using initial values now the particular solution particular solution of this type of differential equation is given by 1 over f of d multiplied by 2 u t so here we have 1 over d square plus 2d plus 10 multiplied 2 u t so we have e to the power here we have e to the bar 0t multiply u t and if we put here 0 it is here d t or we can put 0 and in place of d we can also put 0 so we have 2 over 0 is square 0 plus 2 multiplied by 0 plus 10 multiplied by so it is equal to 2 over 10 which is 1 over 5 so this is the particular integral yp is equal to 1 over 5 now we have in the c part we have to find the total solution and so total solution y t is equal to yc plus yp.

04:10 So here we have yt equal to ycc1, e to t2 minus t, cos 3 t plus 5 u t plus 1 over 5.

04:27 And if we differentiate this equation we have y -dash t equal to we have c1 multiple by minus e to t minus t minus t this will be 3 sign 3 t plus 5 it is 3 sign 3 t plus 5 plus 5 plus cos 3 t plus 5 so and in the equation we are given y 0 equal to 0 y 0 equal to 4 so if we put y 0 equal to 4 in this equation so we have 4 is equal to c1 cost 5 because t will be 0 so c1 cost 5 plus 1 over 5 so we have c1 cost 5 as 19 over 5 similarly in the question we are given y dash 0 equal to 7 so in this equation we put 7 is equal to 7 is equal to in place of t we put 0 so 7 is equal to c1 multiplied by 3 sine 5 plus cos 5.

05:54 So here we have here 7 is equal to 3 .c1 sine 5 plus c1 cos 5...

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