Differential Equations - 40 - Variation of Parameters using Wronskian

Solving y''+y=tan(t) using the Wronskian

[vincentanalik]

Thank you so much for all the videos on this series. They really helped me a lot. I truly appreciate your time and effort.

[kingbeauregard]

The thing I like about the Wronskian is that we've already seen it, more or less, with Cramer's Rule for solving a system of linear equations. For who don't remember it, that's the one where you make a matrix of the coefficients of your x, y, z, etc terms, and take its determinant. Then you take that matrix, replace each column of coefficients with the constants from the other side of the equals sign, take the determinant of THAT matrix, and then divide by the original matrix's determinant.

Well, what we're doing here is Cramer's rule, but applied to functions rather than coefficients and constants. As with linear equations, you first calculate a determinant of a matrix; but then you do a column swap. What you swap into each column in turn is this matrix: [0, 0, 0, ..., f(t)]. Calculate the determinant, divide by the original matrix's determinant, and that will get you all the u-primes. So you see how all of that is Cramer's Rule. One last step to solve this thing: after you've got your u-primes, you'll need to integrate them and then you can assemble the complete solution.

[charlesdingus]

Thank you so much, I had this course in my CS degree. You have helped me a lot.

[Nikhilx]

Thank you so much sir for putting so much efforts in making this series. It has seriously helped me a lot! I really appreciate your efforts!

[jtcarney]

Just made it so easy and simple to understand. Thank you.

[brunomendes]

You know, your channel's name, it's exactly me in life.

[pavankumarachanti]

thank you .thank you so very much . this series really helped me .

[hrushikeshravuri]

Pls come back with more lectures.whats your name?
Do you joined in any educational platforms like udemy and all .Love to learn more from you

[MegaAmged]

Word can describe how I am grateful to you after, and I am also feeling sorry for you because you were not allowed to use this great method LOL

[JoshuaYero-dtjj]

How exactly did you get Y1 and Y2? Are they based off tan (t) ?

[wyvernmonarch]

Can you prove, without complex numbers, that the Taylor series of sin and cos centered at 0 equal the sin and cos function as the number of terms approaches infinity

[joshualimkl]

What if you switch the position of y1 and y2 and their derivatives as well in the wronskian matrix? Does it yield different answer?