❖ Variation of Parameters to Solve a Differential Equation (Second Order) Example 2 ❖

Variation of Parameters for Second-Order Differential Equations (Example 2)

In this video, we'll solve the differential equation y'' + y = 1/(x + 1) using the method of variation of parameters. In this example, we'll walk through each step of finding the solution. Since the integral involved is not straightforward, we'll express our solution using definite integrals rather than finding a simplified closed form. This is a great example of how variation of parameters can be used to tackle more complex differential equations.

What You Will Learn:
How to apply variation of parameters to solve a second-order differential equation.
Understanding how to handle non-homogeneous terms in the equation.
Expressing solutions using definite integrals when the integral is not easily solvable.
Tips for setting up and solving more complex DEs using variation of parameters.
📚 Check out my book: 1001 Calculus Problems for Dummies for more practice!

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[Anonyfagful]

Holy shit! You posted this the day before my midterm involving variation of parameters! What a fortunate coincidence!

[robertstevens]

Could you provide some examples using Wronksian, and undetermined coefficients?

[valeriechan]

another way of finding y is to solve the characteristic equation. in this case we have r^2 + 0r + 1 = 0. Use quadratic formula to solve for r, you get r = +/- i. Recall that if you have a complex root of r = a +/- bi, you get the form y = e^(ax) (c1cos(bx) + c2sin(bx). in this case, we have a = 0 and b = 1. so you get y = e^(0x) (c1cos(x) + c2sin(x)) as your complementary solution. e^(0x) = 1 so your complementary solution is y = c1cos(x) + c2sin(x). then use cos(x) and sin(x) as your y1, y2

[Fldman]

Two words...

Pop. Filter.
Great vids though very helpful

[amandakorant]

Please post some examples on particular solutions for systems involving springs and how to solve them with eigenvalues and eigenvectors!

[raaedk]

Went to see if you still make videos, not disappointed. Good job!

[chrisrichards]

I know this works but the bulk of these problems require the use of Wronskians and the form u'=-y2 * g(t)/W and v'= y1 * g(t)/W, the equations are usually intended to not work out cleanly

[MrYusufgovani]

Patrick! Very Good videos Your explaination is excellent.

[MrAussiePvP]

can someone please link me to the video where he explains that + or - i is equal to [cos(x), sin(x)]. TIA

[Nanski]

I'm stuck at the very beginning, 0:48 can anyone explain to me how do we know that a solution of the form y = e^(rx) exists?

I know that when we have ay'' +by' + cy = 0 then if the roots are distinct, the general solution is y = c1e^(r1x) + c2e^(r2x)
But here we only have ay'' + cy = 0

[GButtersX]

Dang where was this when I needed it a month ago?

[burningoyster]

At 2:06 you talk about having a video about why when r equals i, the solutions are sin(x) and cosine(x). Where can I find this video?

[MiamiGeneralContractor]

thanks, you videos are so helpful but why did not you choose a easier end example lol. 

[hananurrehman]

oh great!
My lecturer taught me the Wronksian method
guess i'll have to stick to that

[kelcamer]

How do you get to the system of equations?  I understand the assumption is y = v1y1 + v2y2 but when you take the derivative of that, why does the u1y' + u2y' go away, and how do you get two solutions?

[illbebak]

Which of your videos should i watch to know why did we use Sin & Cos (The stuff u said that you've explained in a previous vid ?

[RedHaloManiac]

Theres a faster way to get v1 and v2 without doing all the substitution and proof work. You can do and for its much faster that doing substitution and all that

[davidkim]

couldn't you have just used an auxiliary equation by turning y''+ y = 0 into r^2 + 0r + 1 = 0? whats up with the y = e^rx? You didn't use this method in your first example either.

[szhou]

can you use partial fraction decomp to integrate the v1 and v2?

[yairkeller]

at 9:10 i think you can do integral in seprate part