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9.1E: Introduction to Linear Higher Order Equations (Exercises)

( \newcommand{\kernel}{\mathrm{null}\,}\)

Q9.1.1

1. Verify that the given function is the solution of the initial value problem.

  1. x3y3x2y+6xy6y=24x,y(1)=0, y(1)=0,y(1)=0 ;y=6x8x23x3+1x
  2. y1xyy+1xy=x24x4,y(1)=32,y(1)=12,y(1)=1 ;y=x+12x
  3. xyyxy+y=x2,y(1)=2,y(1)=5,y(1)=1 ;y=x22+2e(x1)e(x1)+4x
  4. 4x3y+4x2y5xy+2y=30x2,y(1)=5,y(1)=172 ;y(1)=634;y=2x2lnxx1/2+2x1/2+4x2
  5. x4y(4)4x3y+12x2y24xy+24y=6x4,y(1)=2 ;y(1)=9,y(1)=27,y(1)=52 ;y=x4lnx+x2x2+3x34x4
  6. xy(4)y4xy+4y=96x2,y(1)=5,y(1)=24 ;y(1)=36;y(1)=48;y=912x+6x28x3

2. Solve the initial value problem

x3yx2y2xy+6y=0,y(1)=4,y(1)=14,y(1)=20. HINT: See Example 9.1.1.

3. Solve the initial value problem

y(4)+y7yy+6y=0,y(0)=5,y(0)=6,y(0)=10,y(0)36. HINT: See Example 9.1.2.

4. Find solutions y1, y2, …, yn of the equation y(n)=0 that satisfy the initial conditions

y(j)i(x0)={0,ji1,[5pt]1,j=i1,1in.

5.

  1. Verify that the function y=c1x3+c2x2+c3x satisfies x3yx2y2xy+6y=0 if c1, c2, and c3 are constants.
  2. Use (a) to find solutions y1, y2, and y3 of (A) such that y1(1)=1,y1(1)=0,y1(1)=0[5pt]y2(1)=0,y2(1)=1,y2(1)=0[5pt]y3(1)=0,y3(1)=0,y3(1)=1.
  3. Use (b) to find the solution of (A) such that y(1)=k0,y(1)=k1,y(1)=k2.

6. Verify that the given functions are solutions of the given equation, and show that they form a fundamental set of solutions of the equation on any interval on which the equation is normal.

  1. y+yyy=0;{ex,ex,xex}
  2. y3y+7y5y=0;{ex,excos2x,exsin2x}.
  3. xyyxy+y=0;{ex,ex,x}
  4. x2y+2xy(x2+2)y=0;{ex/x,ex/x,1}
  5. (x22x+2)yx2y+2xy2y=0;{x,x2,ex}
  6. (2x1)y(4)4xy+(52x)y+4xy4y=0;{x,ex,ex,e2x}
  7. xy(4)y4xy+4y=0;{1,x2,e2x,e2x}

7. Find the Wronskian W of a set of three solutions of y+2xy+exyy=0, given that W(0)=2.

8. Find the Wronskian W of a set of four solutions of y(4)+(tanx)y+x2y+2xy=0, given that W(π/4)=K.

9.

  1. Evaluate the Wronskian W {ex,xex,x2ex}. Evaluate W(0).
  2. Verify that y1, y2, and y3 satisfy y3y+3yy=0.
  3. Use W(0) from (a) and Abel’s formula to calculate W(x).
  4. What is the general solution of (A)?

10. Compute the Wronskian of the given set of functions.

  1. {1,ex,ex}
  2. {ex,exsinx,excosx}
  3. {2,x+1,x2+2}
  4. x,xlnx,1/x}
  5. {1,x,x22!,x33!,,xnn!}
  6. {ex,ex,x}
  7. {ex/x,ex/x,1}
  8. {x,x2,ex}
  9. {x,x3,1/x,1/x2}
  10. {ex,ex,x,e2x}
  11. {e2x,e2x,1,x2}

11. Suppose Ly=0 is normal on (a,b) and x0 is in (a,b). Use Theorem 9.1.1 to show that y0 is the only solution of the initial value problem Ly=0,y(x0)=0,y(x0)=0,,y(n1)(x0)=0, on (a,b).

12. Prove: If y1, y2, …, yn are solutions of Ly=0 and the functions zi=nj=1aijyj,1in, form a fundamental set of solutions of Ly=0, then so do y1, y2, …, yn.

13. Prove: If y=c1y1+c2y2++ckyk+yp is a solution of a linear equation Ly=F for every choice of the constants c1, c2,…, ck, then Lyi=0 for 1ik.

14. Suppose Ly=0 is normal on (a,b) and let x0 be in (a,b). For 1in, let yi be the solution of the initial value problem Lyi=0,y(j)i(x0)={0,ji1,1,j=i1,1in, where x0 is an arbitrary point in (a,b). Show that any solution of Ly=0 on (a,b), can be written as y=c1y1+c2y2++cnyn, with cj=y(j1)(x0).

15. Suppose {y1,y2,,yn} is a fundamental set of solutions of  P0(x)y(n)+P1(x)y(n1)++Pn(x)y=0 on (a,b), and let z1=a11y1+a12y2++a1nynz2=a21y1+a22y2++a2nynz11y1+a2y2++anyn=bzn=an1y1+an2y2++annyn, where the {aij} are constants. Show that {z1,z2,,zn} is a fundamental set of solutions of (A) if and only if the determinant |a11a12a1na21a22a2nan1an2ann| is nonzero. HINT: The determinant of a product of n×n matrices equals the product of the determinants.

16. Show that {y1,y2,,yn} is linearly dependent on (a,b) if and only if at least one of the functions y1, y2, …, yn can be written as a linear combination of the others on (a,b).

Q9.1.2

Take the following as a hint in Exercises 9.1.17-9.1.19:

By the definition of determinant, |a11a12a1na21a22a2nan1an2ann|=±a1i1a2i2,,anin, where the sum is over all permutations (ii,i2,,in) of (1,2,,n) and the choice of + or in each term depends only on the permutation associated with that term.

17. Prove: If A(u1,u2,,un)=|a11a12a1na21a22a2nan1,1an1,2an1,nu1u2un|, then A(u1+v1,u2+v2,,un+vn)=A(u1,u2,,un)+A(v1,v2,,vn).

18. Let F=|f11f12f1nf21f22f2nfn1fn2fnn|, where fij(1i,jn) is differentiable. Show that F=F1+F2++Fn, where Fi is the determinant obtained by differentiating the ith row of F.

19. Use Exercise 9.1.18 to show that if W is the Wronskian of the n-times differentiable functions y1, y2, …, yn, then

W=|y1y2yny1y2yny(n2)1y(n2)2y(n2)ny(n)1y(n)2y(n)n|.

Q9.1.3

20. Use Exercises 9.1.17 and 9.1.19 to show that if W is the Wronskian of solutions {y1,y2,,yn} of the normal equation P0(x)y(n)+P1(x)y(n1)++Pn(x)y=0, then W=P1W/P0. Derive Abel’s formula (Equation 9.1.15) from this.

21. Prove Theorem 9.1.6.

22. Prove Theorem 9.1.7.

23. Show that if the Wronskian of the n-times continuously differentiable functions {y1,y2,,yn} has no zeros in (a,b), then the differential equation obtained by expanding the determinant |yy1y2ynyy1y2yny(n)y(n)1y(n)2y(n)n|=0, in cofactors of its first column is normal and has {y1,y2,,yn} as a fundamental set of solutions on (a,b).

24. Use the method suggested by Exercise 9.1.23 to find a linear homogeneous equation such that the given set of functions is a fundamental set of solutions on intervals on which the Wronskian of the set has no zeros.

  1. {x,x21,x2+1}
  2. {ex,ex,x}
  3. {ex,xex,1}
  4. {x,x2,ex}
  5. {x,x2,1/x}
  6. {x+1,ex,e3x}
  7. {x,x3,1/x,1/x2}
  8. {x,xlnx,1/x,x2}
  9. {ex,ex,x,e2x}
  10. {e2x,e2x,1,x2}

This page titled 9.1E: Introduction to Linear Higher Order Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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