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Mathematics LibreTexts

7.5E: Regular Singular Points Euler Equations (Exercises)

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Q7.4.1

In Exercises 7.4.1-7.4.18 find the general solution of the given Euler equation on (0,).

1. x2y+7xy+8y=0

2. x2y7xy+7y=0

3. x2yxy+y=0

4. x2y+5xy+4y=0

5. x2y+xy+y=0

6. x2y3xy+13y=0

7. x2y+3xy3y=0

8. 12x2y5xy+6y=0

9. 4x2y+8xy+y=0

10. 3x2yxy+y=0

11. 2x2y3xy+2y=0

12. x2y+3xy+5y=0

13. 9x2y+15xy+y=0

14. x2yxy+10y=0

15. x2y6y=0

16. 2x2y+3xyy=0

17. x2y3xy+4y=0

18. 2x2y+10xy+9y=0

Q7.4.2

19.

  1. Adapt the proof of Theorem 7.4.3 to show that y=y(x) satisfies the Euler equation ax2y+bxy+cy=0 on (,0) if and only if Y(t)=y(et) ad2Ydt2+(ba)dYdt+cY=0. on (,).
  2. Use (a) to show that the general solution of Equation A on (,0) is y=c1|x|r1+c2|x|r2 if r1 and r2 are distinct real numbers; y=|x|r1(c1+c2ln|x|) if r1=r2y=|x|λ[c1cos(ωln|x|)+c2sin(ωln|x|)] if r1,r2=λ±iω with ω>0.

20. Use reduction of order to show that if

ar(r1)+br+c=0

has a repeated root r1 then y=xr1(c1+c2lnx) is the general solution of

ax2y+bxy+cy=0

on (0,).

21. A nontrivial solution of

P0(x)y+P1(x)y+P2(x)y=0

is said to be oscillatory on an interval (a,b) if it has infinitely many zeros on (a,b). Otherwise y is said to be nonoscillatory on (a,b). Show that the equation

x2y+ky=0(k=constant)

has oscillatory solutions on (0,) if and only if k>1/4.

22. In Example 7.4.2 we saw that x0=1 and x0=1 are regular singular points of Legendre’s equation

(1x2)y2xy+α(α+1)y=0.

  1. Introduce the new variables t=x1 and Y(t)=y(t+1), and show that y is a solution of (A) if and only if Y is a solution of t(2+t)d2Ydt2+2(1+t)dYdtα(α+1)Y=0, which has a regular singular point at t0=0.
  2. Introduce the new variables t=x+1 and Y(t)=y(t1), and show that y is a solution of (A) if and only if Y is a solution of t(2t)d2Ydt2+2(1t)dYdt+α(α+1)Y=0, which has a regular singular point at t0=0.

23. Let P0,P1, and P2 be polynomials with no common factor, and suppose x00 is a singular point of

P0(x)y+P1(x)y+P2(x)y=0.

Let t=xx0 and Y(t)=y(t+x0).
  1. Show that y is a solution of (A) if and only if Y is a solution of R0(t)d2Ydt2+R1(t)dYdt+R2(t)Y=0. where Ri(t)=Pi(t+x0),i=0,1,2.
  2. Show that R0, R1, and R2 are polynomials in t with no common factors, and R0(0)=0; thus, t0=0 is a singular point of (B).

This page titled 7.5E: Regular Singular Points Euler Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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