Kenneth Kuttler

634 CHAPTER 32. SOLUTIONS NEAR A REGULAR SINGULAR POINT

4. In the above problem, suppose pk (x) and pl (x) are solutions, to the equations corre-sponding to n = k, l respectively. Show that∫ 1

−1pk (x) pl (x)dx = 0

Thus this gives an example of a collection of orthogonal polynomials.

5. The Legendre polynomials are given in the above problem but one multiplies by aconstant so that the result satisfies pn (1) = 1. The purpose of this problem is to findthe constant. Hint: Use the Leibniz formula on

(x2−1

)n= (x−1)n (x+1)n.

6. The equation(1− x2

)y′′ − xy′ + n2y = 0 is called the Chebychev equation. Find

solutions to this equation. That is, specify a recurrence relation and two solutions.Explain why there exist polynomial solutions to this equation. Hint: You just lookfor power series solutions.

7. The equation(1− x2

)y′′−3xy′+n(n+2)y = 0 is also called the Chebychev equa-

tion. Find solutions to this equation. That is, specify a recurrence relation and twosolutions. Explain why there exist polynomial solutions to this equation. Hint: Youjust look for power series solutions.

8. Specify two solutions to the following differential equation by determining a recur-rence relation and then describing how to obtain two solutions. Hint: You just lookfor power series solutions.

(a) y′′(x2 +1

)+5xy′+2y = 0.

(b) y′′(x2 +1

)+ xy′+3y = 0.

)+7xy′+4y = 0.

(d) y′′(1−3x2

)+6xy′+4y = 0.

(e) y′′−5x2y′−4xy = 0.

(f) y′′−2x2y′− xy = 0.

(g) y′′+ x2y′+2xy = 0.

(h) y′′−3x2y′− xy = 0.

(i) y′′+2x2y′−4xy = 0.

9. Find the solution to the initial value problem y′′+ sin(x)y′+ cos(3x)y = 0 alongwith the initial conditions y(0) = 1,y′ (0) =−1. You just need to find the first termsof the power series solution up to x4.

10. Find the solution to the initial value problem y′′+ tan(2x)y′+ cos(3x)y = 0 alongwith the initial conditions y(0) =−1,y′ (0) = 2. You just need to find the first termsof the power series solution up to x4.

11. Find the solution to the initial value problem y′′+ tan(5x)y′+ sec(3x)y = 0 alongwith the initial conditions y(0) =−2,y′ (0) = 3. You just need to find the first termsof the power series solution up to x4.

12. Find the general solution to the following Euler equations.

(a) y′′x2−3y′x+3y = 0.

(b) y′′x2 +4y′x−4y = 0.

(d) y′′x2 +6y′x+6y = 0.

(e) y′′x2 +4y′x−4y = 0.

(f) y′′x2−3y′x+4y = 0.

63410.11.12.CHAPTER 32. SOLUTIONS NEAR A REGULAR SINGULAR POINT. In the above problem, suppose p; (x) and p; (x) are solutions, to the equations corre-sponding ton = k,/ respectively. Show that[ Px (x) pi(x)dx =0Thus this gives an example of a collection of orthogonal polynomials.. The Legendre polynomials are given in the above problem but one multiplies by aconstant so that the result satisfies p, (1) = 1. The purpose of this problem is to findthe constant. Hint: Use the Leibniz formula on (x? — 1)" = (x—1)" (x+1)".. The equation (1 — x”) y’ — xy’ +n’*y =0 is called the Chebychev equation. Findsolutions to this equation. That is, specify a recurrence relation and two solutions.Explain why there exist polynomial solutions to this equation. Hint: You just lookfor power series solutions.. The equation (1 — x?) y” — 3xy’ +n(n+2)y = 0 is also called the Chebychev equa-tion. Find solutions to this equation. That is, specify a recurrence relation and twosolutions. Explain why there exist polynomial solutions to this equation. Hint: Youjust look for power series solutions.. Specify two solutions to the following differential equation by determining a recur-rence relation and then describing how to obtain two solutions. Hint: You just lookfor power series solutions.M (2x ) + Sxy’ + 2y =0. (f) y" —2x*y! —xy = 0.(a) y +1x +1) 4+2y'+3y =0.+1(b) y”J}; (g) y +x°y’ + 2xy =0.LN NN(c) y" (x ) + Txy’ + 4y = 0. ;(d) y" (1—3x") + Oxy! +4y =0. (h) yh Sry! ay = 0.(e) y" —5x°y! —4xy = 0. (i) y’ +2x°y/ — 4xy =0.. Find the solution to the initial value problem y” + sin (x) y’ + cos (3x) y = 0 alongwith the initial conditions y(0) = 1, y’ (0) = —1. You just need to find the first termsof the power series solution up to x*.Find the solution to the initial value problem y” + tan (2x) y’ + cos (3x) y = 0 alongwith the initial conditions y(0) = —1,y’ (0) = 2. You just need to find the first termsof the power series solution up to x*.Find the solution to the initial value problem y” + tan (5x) y’ + sec (3x) y = 0 alongwith the initial conditions y(0) = —2,y’ (0) = 3. You just need to find the first termsof the power series solution up to x*.Find the general solution to the following Euler equations.(a) yx? —3y'x+3y =0. (d) yx? + 6y'x+ 6y = 0.(b) y"x* +4y’x—4y =0. (e) yx? + 4y'x—4y =0.(c) yx? + 2y'x—6y =0. (f) y"x? — 3y'x+4y =0.