Problem 1. (10 points) Find the general solution of the differential equation (d^2y)/(dx^2) + 3(

Step 1: The given differential equation is a non-homogeneous linear differential equation with c

Problem 1. (10 points) Find the general solution of the...

Transcript

00:01 Dear students, here we have a differential equation which is non -homogeneous.

00:06 So we have to find the general solution of this equation by the undeterminate, the method of undeterminate coefficient.

00:15 So we know that we can write our differential operator d upon d x in the form of d as d is our differential operator.

00:24 So by writing our equation in the form of our differential operator we'll get d square y plus 2dy plus 5y equals to 6 sine x 2x into plus 7 cross 2x let's give this our the name of our first equation now we can take the auxiliary equation from our first equation as our auxiliary equation will be it will be m square so we can write our auxiliary equation in the form of f into m equals to zero as it is in the form of f of d so we'll write it as m square plus 2m plus 5 equals to 0 so we can solve our quality equation by quality formula as it will be m equals to minus which is minus of 2 plus minus square root of b square which is 2 square which will we will get 4 into minus 4 into a c we've got 4 into 5 we'll get minus of 20 upon 2 so by further simplifying it will get minus 2 plus minus square root of minus 16 upon 2 so then we just we know that the square root of minus 1 is iota and the square root of 16 is 4 so we'll get 4 iota upon 2 by dividing upon 2 we'll get minus 1 plus minus of 2 aorta so this is our root which is in complex form as minus 1 is our real term and 2 aorta is our main letter so we can write our complete elementary function or solution as c1 e to the power minus x so we won't write it as c1 so write it as e to the power minus x into c1 cause into we have two of it as i we have two of it as two x in the angle plus c2 sine 2x so this will be our complementary function now for our particular integral we'll write it as or y into p will be as you've got our right -hand side as the function of sine and cause for our function of sign and cause we will take our y p as a sine 2x plus b cause 2x as you've got different values of our yc and yp.

04:14 So if we, our yc matches our yp, so it will change our yp just by multiplying with x.

04:27 So this time it didn't match...

Question

Problem 1. (10 points) Find the general solution of the differential equation \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 1 + 5\sin x.

Question

Problem 1. (10 points) Find the general solution of the differential equation \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 1 + 5\sin x.

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