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4.2E: Introduction to Linear Higher Order Equations (Exercises)

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

Q4.2.1

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,ji1,1,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)=0y2(1)=0,y2(1)=1,y2(1)=0y3(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}

Q4.2.2

11. 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).

12. Use the method suggested by Exercise 4.2.11 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 4.2E: 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 Zoya Kravets.

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