Question

(Legendre polynomials). Let $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$, and set $y_n = \frac{d^n}{dx^n}(1-x^2)^n$. Show that $y_n$ is a solution of the differential equation $(1 - x^2) y'' - 2x y' + n(n+1) y = 0$. Hint: find a first order differential equation which is satisfied by $u = (1 - x^2)^n$, then differentiate this $n + 1$ times.

Name: legendre polynomials let n in mathbbn0 mathbbn cup 0 and set yn fracdndxn1 x2n show that yn is a solution of the differential equation 1 x2 y 2x y nn1 y 0 hint find a first order differen 43691
Uploaded: 2020-05-28T21:26:28-08:00
Duration: 4 min 12 s
Channel: Alexandra Embry
Description: legendre polynomials let n in mathbbn0 mathbbn cup 0 and set yn fracdndxn1 x2n show that yn is a solution of the differential equation 1 x2 y 2x y nn1 y 0 hint find a first order differen 43691

          (Legendre polynomials). Let $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$, and set $y_n = \frac{d^n}{dx^n}(1-x^2)^n$.
Show that $y_n$ is a solution of the differential equation $(1 - x^2) y'' - 2x y' + n(n+1) y = 0$.
Hint: find a first order differential equation which is satisfied by $u = (1 - x^2)^n$, then differentiate this $n + 1$ times.

$legendre polynomials let n in mathbbn0 mathbbn cup 0 and set yn fracdndxn1 x2n show that yn is a solution of the differential equation 1 x2 y 2x y nn1 y 0 hint find a first order differen 43691$

Added by Ashley G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Alexandra Embry

05/28/2020

Step 1

Then, we find the first derivative of $u$ with respect to $x$: $$u' = \frac{du}{dx} = n(1-x^2)^{n-1}(-2x) = -2nx(1-x^2)^{n-1}$$ We can rewrite this as: $$u' = \frac{-2nx}{1-x^2}(1-x^2)^n = \frac{-2nx}{1-x^2}u$$ Thus, we have the first order differential Show more…

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(Legendre polynomials). Let $n \in \mathbb{N}_0 = \mathbb{N} \cup \{0\}$, and set $y_n = \frac{d^n}{dx^n}(1-x^2)^n$. Show that $y_n$ is a solution of the differential equation $(1 - x^2) y'' - 2x y' + n(n+1) y = 0$. Hint: find a first order differential equation which is satisfied by $u = (1 - x^2)^n$, then differentiate this $n + 1$ times.