Use the variation-of-parameters method to find the general solution to the given differential eq

Use the variation-of-parameters method to find the general...

Key Concepts

Wronskian

The Wronskian is a determinant used to verify the linear independence of the fundamental solutions of a differential equation. In the variation of parameters method, the Wronskian plays a key role in constructing the formulas for the parameter functions that are integrated to find the particular solution.

Homogeneous Differential Equation

This concept involves solving the associated homogeneous version of the given non-homogeneous differential equation. The homogeneous equation is found by setting the non-homogeneous term (the right?hand side) to zero. The solution of the homogeneous equation yields a set of fundamental solutions which form the complementary solution to the original differential equation.

Fundamental Set of Solutions

These are linearly independent solutions to the homogeneous differential equation. They are typically found by solving the characteristic equation when the differential equation has constant coefficients. The general homogeneous solution is expressed as a linear combination of these fundamental solutions.

Variation of Parameters

This is a method used to find a particular solution to a non-homogeneous linear differential equation. The method involves using the fundamental set of solutions from the homogeneous equation to construct an ansatz where the constants are replaced by functions. These functions are then determined by imposing conditions that simplify the system, allowing one to integrate and obtain the particular solution.

Transcript

00:01 So to start this problem, we're first going to solve the corresponding homogeneous problem by first solving the auxiliary equation.

00:07 That's p of r is equal to r squared minus 4 is equal to 0.

00:11 So then r is going to be plus or minus 2.

00:14 So our complementary solution, yc of x, is going to be equal to c1 e to the 2x plus c2 e to the negative 2x.

00:25 So our y1 is going to be e to the 2x, and then our y2 is going to be e to the negative 2x.

00:31 2x, our particular solution is going to take the form u1y1 plus u2, y2, where u1 and u2 are going to satisfy the following two equations.

00:44 We're going to have y1, so e to the 2x times u1 prime, plus e to the negative 2x times u2 prime equals 0, and then y1 prime times u1 prime plus uy2 prime times u2 prime will be equal to, and then that's going to be equal to our right -hand side of the equation.

01:15 So 8 over e to the 2x plus 1.

01:21 Okay.

01:22 Next, we're going to use kramer's rule.

01:25 So kramer's rule tells us that u1 prime is going to be equal to w1 over w, and then u2 prime is going to be equal to w2 over w, where here, w is going to be the determinant of the ron, or it's going to be the ron scan.

01:43 So that's, we have 2e to the 2x, e to the 2x, and then e to the negative 2x, and negative 2e to the negative 2x.

01:52 So the determinant of that, okay? so when we multiply this by this, we just get negative 2, and then this times this, then we'll get minus 2 as well.

02:06 So this is going to be equal to negative 4.

02:11 Next, we need to find w1.

02:13 So w1 is going to look like the ronskin, except we substitute this into the first column.

02:21 So we have 8 over e to the 2x plus 1, 0, e to the negative 2x, negative 2, negative 2x.

02:33 Okay.

02:34 And then now this times this is 0.

02:37 And then here we have negative e to the negative 2x over, or sorry, 8 times that, 8 e to negative 2x divided by e to the 2x plus 1.

02:51 This can actually simplify if we multiply the top and bottom by e to the 2x, we can get negative 8 over e to the 4x plus e to 2x.

03:03 Actually, let's just leave it like this for now.

03:07 So w2, then we're, we're going to be.

03:12 Going to leave the first column now this time and then we're going to substitute in our second column.

03:24 Okay, and then zero here.

03:27 So this is going to be equal to 8e to the 2x divided by e to the 2x plus 1.

03:35 And then minus 0.

03:39 So then now are u1 prime, this is equal to again w1 over w.

03:50 So w1 was equal to this here, negative 8, negative 8, e to the negative 2x divided by e to the 2x plus 1, and then we're going to divide all of that by negative 4.

04:08 So u1 prime is going to be equal to.

04:13 Then we'll have 2e to the negative 2x, then e to the 2x plus 1.

04:22 Okay, so to first integrate this, we're going to use this.

04:27 Substitution or use the first substitution that u is going to be equal to e to the 2x so then now d u is equal to here uh 2 e to the 2x d x and then now okay so when we take the integral to substitute so this is going to be equal to and i'm going to notice also um so dx is going to be equal to 1 over 2, or 1 over 2, or 1 over 2, and then e to the 2x is u.

05:22 So it's going to be du divided by 2u like so.

05:28 Okay.

05:33 So when we make in the substitution, so this integral, or this e to the 2x can actually go down as e to the 2x, which we know is u.

05:48 So again, making our substitutions, we have 2 over u times u plus 1.

05:59 That's this.

06:01 And then du divided by 2u.

06:05 That's our dx.

06:07 So now the 2s will cancel out.

06:10 So all we're left with is the integral of 1 over u.

06:15 Squared times u plus 1 du.

06:20 Now let's put the du up here.

06:23 So we can decompose this into partial fractions.

06:27 So doing partial fractions, so we want that 1 over u squared times u plus 1.

06:41 We want this to be equal to.

06:46 So first we have a knot over u plus plus b0 over u squared, and then plus c0 over u plus 1.

07:05 So we need to solve for a not, b not, and c not.

07:09 So let's multiply now everything.

07:14 So here we get a not times u.

07:17 Again we're multiplying everything by u squared times u plus 1.

07:22 So this becomes u times u plus 1 and then plus b not times u plus 1 and then plus c not u squared so let's multiply some more of this out let me scoop this up some here now this right -hand side becomes a not u squared plus a not u plus b not u plus b not u plus b not u plus c 0 u squared...

Please give Ace some feedback