9.1E: Introduction to Linear Higher Order Equations (Exercises)
- Page ID
- 44289
( \newcommand{\kernel}{\mathrm{null}\,}\)
Q9.1.1
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,[5pt]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)=0[5pt]y2(1)=0,y′2(1)=1,y″2(1)=0[5pt]y3(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}
11. Suppose Ly=0 is normal on (a,b) and x0 is in (a,b). Use Theorem 9.1.1 to show 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).
12. Prove: If y1, y2, …, yn are solutions of Ly=0 and the functions zi=n∑j=1aijyj,1≤i≤n, 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 1≤i≤k.
14. Suppose Ly=0 is normal on (a,b) and let x0 be in (a,b). For 1≤i≤n, let yi be the solution of the initial value problem Lyi=0,y(j)i(x0)={0,j≠i−1,1,j=i−1,1≤i≤n, 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(j−1)(x0).
15. Suppose {y1,y2,…,yn} is a fundamental set of solutions of P0(x)y(n)+P1(x)y(n−1)+⋯+Pn(x)y=0 on (a,b), and let z1=a11y1+a12y2+⋯+a1nynz2=a21y1+a22y2+⋯+a2nynz1⋮1y1+a⋮2y2+⋯+a⋮nyn=b⋮zn=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 |a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann| 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, |a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann|=∑±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)=|a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an−1,1an−1,2⋯an−1,nu1u2⋯un|, then A(u1+v1,u2+v2,…,un+vn)=A(u1,u2,…,un)+A(v1,v2,…,vn).
18. Let F=|f11f12⋯f1nf21f22⋯f2n⋮⋮⋱⋮fn1fn2⋯fnn|, where fij(1≤i,j≤n) 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′=|y1y2⋯yny′1y′2⋯y′n⋮⋮⋱⋮y(n−2)1y(n−2)2⋯y(n−2)ny(n)1y(n)2⋯y(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(n−1)+⋯+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 |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).
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.
- {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}