7.1.1: Introduction to Linear Higher Order Equations (Exercises)
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- Oct 9, 2022
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In Exercises 1-6 verify that the given function is the solution of the initial value problem.
1. Verify that y=−6x−8x2−3x3+1x satisfies the IVP:
x3y‴−3x2y″+6xy′−6y=−24x;y(−1)=0,y′(−1)=0,y″(−1)=0
2. Verify that y=x+12x satisfies the IVP:
y‴−1xy″−y′+1xy=x2−4x4,y(1)=32,y′(1)=12,y″(1)=1
3. Verify that y=−x2−2+2e(x−1)−e−(x−1)+4x satisfies the IVP:
xy‴−y″−xy′+y=x2,y(1)=2,y′(1)=5,y″(1)=−1
4. Verify that y=2x2lnx−x1/2+2x−1/2+4x2 satisfies the IVP:
4x3y‴+4x2y″−5xy′+2y=30x2,y(1)=5,y′(1)=172,y″(1)=634
5. Verify that y=x4lnx+x−2x2+3x3−4x4 satisfies the IVP
x4y(4)−4x3y‴+12x2y″−24xy′+24y=6x4,y(1)=−2,y′(1)=−9,y″(1)=−27,y‴(1)=−52
6. Verify that y=9−12x+6x2−8x3 satisfies the IVP:
xy(4)−y‴−4xy″+4y′=96x2,y(1)=−5,y′(1)=−24,y″(1)=−36;y‴(1)=−48
7. Using the solution from Example 7.1.1, solve the initial value problem
x3y‴−x2y″−2xy′+6y=0,y(−1)=−4,y′(−1)=−14,y″(−1)=−20.
8. Using the solution from Example 7.1.2, solve the initial value problem
y(4)+y‴−7y″−y′+6y=0,y(0)=5,y′(0)=−6,y″(0)=10,y‴(0)−36.
9.
- 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.
In Exercises 10-16 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 has continuous coefficients.
10. y‴+y″−y′−y=0;{ex,e−x,xe−x}
11. y‴−3y″+7y′−5y=0;{ex,excos2x,exsin2x}
12. xy‴−y″−xy′+y=0;{ex,e−x,x}
13. x2y‴+2xy″−(x2+2)y=0;{ex/x,e−x/x,1}
14. (x2−2x+2)y‴−x2y″+2xy′−2y=0;{x,x2,ex}
15. (2x−1)y(4)−4xy‴+(5−2x)y″+4xy′−4y=0;{x,ex,e−x,e2x}
16. xy(4)−y‴−4xy′+4y′=0;{1,x2,e2x,e−2x}
In Exercise 17-27 compute the Wronskian of the given set of functions and what your result says about the linear independence of the set.
17. {1,ex,e−x}
18. {ex,exsinx,excosx}
19. {2,x+1,x2+2}
20. {x,xlnx,1/x}
21. {1,x,x22!,x33!,⋯,xnn!}
22. {ex,e−x,x}
23. {ex/x,e−x/x,1}
24. {x,x2,ex}
25. {x,x3,1/x,1/x2}
26. {ex,e−x,x,e2x}
27. {e2x,e−2x,1,x2}
28. Suppose y(n)+p1(x)y(n−1)+⋯+pn(x)y=f(x) has continuous coefficients on (a,b) and x0 is in (a,b). Use Theorem 7.1.1 to prove that y≡0 is the only solution of the initial value problem Ly=0,y(x0)=0,y′(x0)=0,…,y(n−1)(x0)=0, on (a,b).
29. Prove: If y=c1y1+c2y2+⋯+ckyk+yp is a solution of a linear equation y(n)+p1(x)y(n−1)+⋯+pn(x)y=f(x) for every choice of the constants c1, c2,…, ck, then yi is a solution to y(n)+p1(x)y(n−1)+⋯+pn(x)y=0 for 1≤i≤k.
