7.1.1: Introduction to Linear Higher Order Equations (Exercises)

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In Exercises 1-6 verify that the given function is the solution of the initial value problem.

1. Verify that \(y=-6x-8x^2-3x^3 + {1\over x}\) satisfies the IVP:

\(x^3y'''-3x^2y''+6xy'-6y=\dfrac{-24}{ x}; \quad y(-1)=0,\quad y'(-1)=0, \quad y''(-1)=0\)

2. Verify that \(y=x+ \dfrac{1}{2x}\) satisfies the IVP:

\(y'''- \dfrac{1}{x}y''-y'+ \dfrac{1}{x}y= \dfrac{x^2-4}{x^4}, \quad y(1)= \dfrac{3}{2}, \quad y'(1)= \dfrac{1}{2}, \quad y''(1)=1\)

3. Verify that \(y=-x^2-2+2e^{(x-1)}-e^{-(x-1)}+4x\) satisfies the IVP:

\(xy'''-y''-xy'+y=x^2, \quad y(1)=2,\quad y'(1)=5,\quad y''(1)=-1\)

4. Verify that \(y=2x^2\ln x-x^{1/2}+2x^{-1/2}+4x^2\) satisfies the IVP:

\(4x^3y'''+4x^2y''-5xy'+2y=30x^2, \quad y(1)=5,\quad y'(1)= \dfrac{17}{2},\quad y''(1)= \dfrac{63}{4}\)

5. Verify that \(y=x^4\ln x+x-2x^2+3x^3-4x^4\) satisfies the IVP

\(x^4y^{(4)}-4x^3y'''+12x^2y''-24xy'+24y=6x^4, \quad y(1)=-2, \quad y'(1)=-9, \quad y''(1)=-27,\quad y'''(1)=-52\)

6. Verify that \(y=9-12x+6x^2-8x^3\) satisfies the IVP:

\(xy^{(4)}-y'''-4xy''+4y'=96x^2, \quad y(1)=-5,\quad y'(1)=-24, \quad y''(1)=-36; \quad y'''(1)=-48\)

7. Using the solution from Example 7.1.1, solve the initial value problem

\[x^3y'''-x^2 y''-2xy'+6y=0, \quad y(-1)=-4, \quad y'(-1)=-14,\quad y''(-1)=-20.\nonumber \]

8. Using the solution from Example 7.1.2, solve the initial value problem

\[y^{(4)}+y'''-7y''-y'+6y=0, \quad y(0)=5,\quad y'(0)=-6,\quad y''(0)=10,\quad y'''(0)-36.\nonumber \]

Verify that the function \[y=c_1x^3+c_2x^2+{c_3\over x}\nonumber \] satisfies \[x^3 y'''-x^2y''-2xy'+6y=0 \tag{A}\] if \(c_1\), \(c_2\), and \(c_3\) are constants.
Use (a) to find solutions \(y_1\), \(y_2\), and \(y_3\) of (A) such that \[\begin{array}{rl} y_1(1)=1,\quad y_1'(1)=0,\quad y_1''(1)=0 \\[5 pt] y_2(1)=0,\quad y_2'(1)=1,\quad y_2''(1)=0 \\[5 pt] y_3(1)=0,\quad y_3'(1)=0,\quad y_3''(1)=1. \end{array}\nonumber \]

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. \(y'''+y''-y'-y=0; \quad\{e^x,\,e^{-x},\,xe^{-x}\}\)

11. \(y'''-3y''+7y'-5y=0; \quad\{e^x,\,e^x\cos2x,\,e^x\sin2x\}\)

12. \(xy'''-y''-xy'+y=0; \quad \{e^x,\,e^{-x},\,x\}\)

13. \(x^2y'''+2xy''-(x^2+2)y=0; \quad \{e^x/ x,\,e^{-x}/ x,\,1\}\)

14. \((x^2-2x+2)y'''-x^2y''+2xy'-2y=0; \quad \{x,\,x^2,\,e^x\} \)

15. \((2x-1)y^{(4)}-4xy'''+(5-2x)y''+4xy'-4y=0; \quad\{x,\,e^x,\,e^{-x},e^{2x}\}\)

16. \(xy^{(4)}-y'''-4xy'+4y'=0; \quad\{1,x^2,\,e^{2x},\,e^{-2x}\}\)

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. \(\{1,\,e^x,\,e^{-x}\}\)

18. \(\{e^x,\, e^x\sin x,\,e^x\cos x\}\)

19. \(\{2,\,x+1,\,x^2+2\}\)

20. \(\{x,\,x\ln x,\,1/x\}\)

21. \(\{1,\,x,\,{x^2\over2!},\, {x^3\over3!}\,,\cdots,\,{x^n\over n!}\}\)

22. \(\{e^x,\,e^{-x},\,x\}\)

23. \(\{e^x/x,\,e^{-x}/x,\,1\}\)

24. \(\{x,\,x^2,\,e^x\}\)

25. \(\{x,\,x^3,\,1/x,\,1/x^2\}\)

26. \(\{e^x,\,e^{-x},\,x,\,e^{2x}\}\)

27. \(\{e^{2x},\,e^{-2x},\,1,\,x^2\}\)

28. Suppose \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x) \) has continuous coefficients on \((a,b)\) and \(x_0\) is in \((a,b)\). Use Theorem 7.1.1 to prove that \(y\equiv0\) is the only solution of the initial value problem \[Ly=0, \quad y(x_0)=0,\quad y'(x_0)=0,\dots, y^{(n-1)}(x_0)=0,\nonumber \] on \((a,b)\).

29. Prove: If \[y=c_1y_1+c_2y_2+\cdots+c_ky_k+y_p\nonumber \] is a solution of a linear equation \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=f(x) \) for every choice of the constants \(c_1\), \(c_2\),…, \(c_k\), then \(y_i\) is a solution to \( y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y=0\) for \(1\le i\le k\).