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.
- x3y‴−3x2y″+6xy′−6y=−24x,y(−1)=0, y′(−1)=0,y″(−1)=0 ;y=−6x−8x2−3x3+1x
- y‴−1xy″−y′+1xy=x2−4x4,y(1)=32,y′(1)=12,y″(1)=1 ;y=x+12x
- xy‴−y″−xy′+y=x2,y(1)=2,y′(1)=5,y″(1)=−1 ;y=−x2−2+2e(x−1)−e−(x−1)+4x
- 4x3y‴+4x2y″−5xy′+2y=30x2,y(1)=5,y′(1)=172 ;y″(1)=634;y=2x2lnx−x1/2+2x−1/2+4x2
- x4y(4)−4x3y‴+12x2y″−24xy′+24y=6x4,y(1)=−2 ;y′(1)=−9,y″(1)=−27,y‴(1)=−52 ;y=x4lnx+x−2x2+3x3−4x4
- xy(4)−y‴−4xy″+4y′=96x2,y(1)=−5,y′(1)=−24 ;y″(1)=−36;y‴(1)=−48;y=9−12x+6x2−8x3
2. Solve the initial value problem
x3y‴−x2y″−2xy′+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)+y‴−7y″−y′+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,j≠i−1,1,j=i−1,1≤i≤n.
5.
- Verify that the function y=c1x3+c2x2+c3x satisfies x3y‴−x2y″−2xy′+6y=0 if c1, c2, and c3 are constants.
- Use (a) to find solutions y1, y2, and y3 of (A) such that y1(1)=1,y′1(1)=0,y″1(1)=0y2(1)=0,y′2(1)=1,y″2(1)=0y3(1)=0,y′3(1)=0,y″3(1)=1.
- 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.
- y‴+y″−y′−y=0;{ex,e−x,xe−x}
- y‴−3y″+7y′−5y=0;{ex,excos2x,exsin2x}.
- xy‴−y″−xy′+y=0;{ex,e−x,x}
- x2y‴+2xy″−(x2+2)y=0;{ex/x,e−x/x,1}
- (x2−2x+2)y‴−x2y″+2xy′−2y=0;{x,x2,ex}
- (2x−1)y(4)−4xy‴+(5−2x)y″+4xy′−4y=0;{x,ex,e−x,e2x}
- xy(4)−y‴−4xy′+4y′=0;{1,x2,e2x,e−2x}
7. Find the Wronskian W of a set of three solutions of y‴+2xy″+exy′−y=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.
- Evaluate the Wronskian W {ex,xex,x2ex}. Evaluate W(0).
- Verify that y1, y2, and y3 satisfy y‴−3y″+3y′−y=0.
- Use W(0) from (a) and Abel’s formula to calculate W(x).
- What is the general solution of (A)?
10. Compute the Wronskian of the given set of functions.
- {1,ex,e−x}
- {ex,exsinx,excosx}
- {2,x+1,x2+2}
- x,xlnx,1/x}
- {1,x,x22!,x33!,⋯,xnn!}
- {ex,e−x,x}
- {ex/x,e−x/x,1}
- {x,x2,ex}
- {x,x3,1/x,1/x2}
- {ex,e−x,x,e2x}
- {e2x,e−2x,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 |yy1y2⋯yny′y′1y′2⋯y′n⋮⋮⋮⋱⋮y(n)y(n)1y(n)2⋯y(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.
- {x,x2−1,x2+1}
- {ex,e−x,x}
- {ex,xe−x,1}
- {x,x2,ex}
- {x,x2,1/x}
- {x+1,ex,e3x}
- {x,x3,1/x,1/x2}
- {x,xlnx,1/x,x2}
- {ex,e−x,x,e2x}
- {e2x,e−2x,1,x2}