In this problem you will use variation of parameters to...

In this problem you will use variation of parameters to solve the nonhomogeneous equation y'' + 4y' + 4y = -6e^-2t A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.) B. Write the fundamental solutions for the associated homogeneous equation and their Wronskian. y1 = y2 = W(y1, y2) = C. Compute the following integrals. ? (y1g/W) dt = ? (y2g/W) dt = D. Write the general solution. (Use c1 and c2 for c1 and c2). y = (Note: Your general solution will only be correct if it is a general solution to the differential equation.)

Transcript

00:01 In this problem, we are provided with the differential equation y double prime plus 4 times y prime plus 4 times y equals to negative 6 times e raised to the power negative 2t.

00:15 And we need to solve this using variation of parameters method.

00:19 So first we consider the characteristic equation.

00:22 We have lambda squared plus 4 times lambda plus 4 equals to 0, which can be written as lambda plus 2 the whole squared.

00:31 Equals to 0 which implies that we have lambda to be equal to negative 2 and negative 2.

00:37 So this gives y 1 equals to e raised to the power negative 2 t and y2 equals to t times e raised to the power negative 2 t.

00:47 So now that we have found y 1 and y2 we can find out the wrong skin.

00:51 So we have the wrong skin to be given by the determinant of y 1, y2, y 1 prime, y2 prime.

00:59 So substituting the values we have the determi of e -rase to the power negative 2t, t times e r -rase to the power negative 2t.

01:08 The derivative of e -r -r -t -2 -t is negative 2 -t2 times e -r -r - to the par negative -t, and the derivative of t -t times e -r -r -megative 2t is e -rased to the par negative -2 -t.

01:24 This is obtained by the product rule of differentiation.

01:28 So now let us find the determinant.

01:31 We get e raised to the power negative 4 t times 1 minus 2 t plus 2 t times e raise to the power negative 40 so now let us simplify this we can factor out e raised to the power negative 40 so we get e raised to the power negative 40 times 1 minus 2 t plus 2 t so canceling out negative 2 t and positive 2 t we get e raised to the power and negative 40 so this is the value of the wrong skin.

02:02 Now that if we have found the wrong skin, we can find out u1 and u2.

02:06 We have u1 equals to integral y1g over wrong skin d t.

02:13 So substituting the values we have integral of e raised to the power negative 2t times negative 6 times e raised to the power negative 2t, the whole divided by e raised to the power negative 40 d t.

02:26 So clearly e r...

Question

Please give Ace some feedback