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

Non-homogeneous Differential Equations

A non-homogeneous differential equation includes a forcing term or a non-zero right-hand side, representing an external input or forcing function. The overall solution is then a combination of the homogeneous solution and a particular solution that specifically addresses the non-homogeneity of the equation.

Wronskian

The Wronskian is a determinant constructed from a set of solutions of a differential equation and their derivatives, and it is used to check the linear independence of those solutions. In the variation of parameters method, the Wronskian is essential for computing the unknown functions that modify the homogeneous solutions to form the particular solution.

Variation of Parameters

Variation of parameters is a method used to find a particular solution to a non-homogeneous linear differential equation. Rather than guessing a form for the particular solution, this technique uses the solutions of the corresponding homogeneous equation and introduces unknown functions in place of the constants. These functions are then determined by substituting back into the original differential equation and solving the resulting equations.

Homogeneous Solution

The homogeneous solution involves solving the associated homogeneous differential equation, which is obtained by setting the non-homogeneous part to zero. This usually requires finding the roots of the characteristic polynomial, and the nature of these roots (real, repeated, or complex) determines the form of the general homogeneous solution.

Transcript

00:01 To start solving this problem, first let's find the homogeneous or the complementary function.

00:07 So to do that, we first solve the auxiliary equation.

00:11 P of r is equal to r square and minus 4r plus 5 is equal to 0.

00:17 Let's try to attempt to factor this first.

00:20 So that means we need to look at the factors of 5.

00:23 That equals, or that's just only 1 and 5, which doesn't add up to 4.

00:27 So we're going to need to use quadratic equation here.

00:31 So we're going to have negative b.

00:33 So that's negative negative, negative 4, plus or minus, and then we're going to have b squared, which is 16, minus 4a, c, all over 2a, which is just 2 times 1.

00:46 So this out front is just a 2.

00:49 So you have a 2 plus or minus.

00:51 And then now here we have 16 minus 20, square root of that is going to be equal to square root of negative.

00:59 4, which is equal to 2i, and then we here have a 2 on the bottom.

01:04 So this is going to be plus or minus i.

01:07 That means that our complementary solution is going to be, or y of c of x, is going to be equal to c1e to the negative, or sorry, c1e to the 2t times cosine of t, plus c2e to the 2t, sign of t y1 is going to be equal to then next we need to find our particular solution so let me just move some of this around here first our particular solution is going to take the form y of p is equal to u1 y1 plus u2 y2 where this is y1 here y1 and this part here is y2 2.

02:06 So u1 and you two are going to satisfy the following two equations.

02:10 So it's going to be first y1 so e to the 2t cosine of t and then u1 prime and then plus then e to the 2t sine of t.

02:25 U2 prime is equal to 0 and then we take the derivative of this of y1 so that's going to be equal to.

02:35 So first times derivative second, it's going to be negative e to the 2t, sign of t, and then plus 2e to the t, cosine of t, and then times u1 prime, plus.

02:52 And then here we're going to have e to the 2t cosine of t minus, or sorry, plus 2e to the 2t, sine of t, minus, or sorry, plus 2e to the 2t, sign of t, t, times u2 prime and this is going to be equal to the right hand side e to the 2x tangent x so using kramer's rule we're going to get that u1 prime is going to be equal to w1 over w and u2 prime is going to be equal to w2 over w where these are going to be the determinants of matrices the ws so w is going to be the ronskin so that's going to be equal to.

03:37 So if we take the here, it's going to be e to the negative or negative e to the 2t, sign of t plus 2e to the t, cosine of t.

03:52 Then here we have e to the 2t, cosine of t, plus 2e to the 2t, sine of t.

04:04 Here we just have e to the 2t, cosine of t, and then here we also have e to the 2t, and this one will be sine of t.

04:17 So taking the nironskin, we're going to multiply this by this.

04:24 So that's going to be equal to.

04:26 So we'll have e to the 4t times cosine squared t, and then plus 2e to the 4t, and then plus 2e to the 4t, sign of t.

04:41 Then we're going to take this, multiply that by that, and then subtract it.

04:46 So minus, and then our first term is going to be plus e to the 4t, sine squared t, and then minus 2e to the t, cosine of t.

05:04 Then now we can simplify a little bit more.

05:07 This e to the 4t, we can factor out out of these two terms.

05:11 Actually, we can factor out of all of the terms, but this cosine square t and the sine square t become one.

05:19 So we actually just get e to the 4t times 1 plus 2 sine t plus or 2 sine t minus 2 cosine t.

05:34 Oh, i just realized i miscalculated here.

05:40 This should be a cosine sine.

05:42 I forgot to multiply.

05:44 When i multiply this by this term, i forgot to multiply the cosine and sign.

05:49 So i get cosine t, t, sine t.

05:55 And then also here, again, i forgot to multiply this cosine and sign.

06:00 So i get cosine t, sine t also.

06:05 Cosine t, sine t.

06:07 Now these two terms will cancel out.

06:10 And again, so this, and then by pythagorean identity, this just becomes e to the 4t.

06:18 Now w1 and w2.

06:21 So w1 first, we're going to calculate by substituting the first column.

06:29 Okay, so the first column is going to be replaced with 0, e to the 2x, tangent x, and then we put in our second column, which is going to be equal to e to the 2t cosine t.

06:46 So e to the 2t cosine t and then plus 2e to the 2t sine t and then here we had e to the 2t sine of t so calculating this ronskin or it's the calculating this determinant so first we're going to get this times this is zero and then minus this times this oops there should be uh it should be all oops, i just realized i have t's instead of x...

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