9.1E: Introduction to Linear Higher Order Equations (Exercises)

Q9.1.1

1. Verify that the given function is the solution of the initial value problem.

1. \(x^3y'''-3x^2y''+6xy'-6y=\dfrac{-24}{ x}, \quad y(-1)=0\), \(y'(-1)=0, \quad y''(-1)=0\) ;\(y=-6x-8x^2-3x^3 + {1\over x}\)
2. \(y'''- \dfrac{1}{x}y''-y'+ \dfrac{1}{x}y= \dfrac{x^2-4}{x^4}, \quad y(1)= \dfrac{3}{2}, \quad y'(1)= \dfrac{1}{2}, y''(1)=1\) ;\(y=x+ \dfrac{1}{2x}\)
3. \(xy'''-y''-xy'+y=x^2, \quad y(1)=2,\quad y'(1)=5,\quad y''(1)=-1\) ;\(y=-x^2-2+2e^{(x-1)}-e^{-(x-1)}+4x\)
4. \(4x^3y'''+4x^2y''-5xy'+2y=30x^2, \quad y(1)=5,\quad y'(1)= \dfrac{17}{2}\) ;\(y''(1)= \dfrac{63}{4};\quad y=2x^2\ln x-x^{1/2}+2x^{-1/2}+4x^2\)
5. \(x^4y^{(4)}-4x^3y'''+12x^2y''-24xy'+24y=6x^4, \quad y(1)=-2\) ;\(y'(1)=-9, \quad y''(1)=-27,\quad y'''(1)=-52\) ;\(y=x^4\ln x+x-2x^2+3x^3-4x^4\)
6. \(xy^{(4)}-y'''-4xy''+4y'=96x^2, \quad y(1)=-5,\quad y'(1)=-24\) ;\(y''(1)=-36; \quad y'''(1)=-48;\quad y=9-12x+6x^2-8x^3\)

2. Solve the initial value problem

\[x^3y'''-x^2 y''-2xy'+6y=0, \quad y(-1)=-4, \quad y'(-1)=-14,\quad y''(-1)=-20.\nonumber \] HINT: See Example 9.1.1.

3. Solve the initial value problem

\[y^{(4)}+y'''-7y''-y'+6y=0, \quad y(0)=5,\quad y'(0)=-6,\quad y''(0)=10,\quad y'''(0)-36.\nonumber \] HINT: See Example 9.1.2.

4. Find solutions \(y_1\), \(y_2\), …, \(y_n\) of the equation \(y^{(n)}=0\) that satisfy the initial conditions

\[y_i^{(j)}(x_0)=\left\{\begin{array}{cl} 0,&j\ne i-1,\\[4pt][5 pt] 1,&j=i-1,\end{array}\right.\; 1\le i\le n.\nonumber \]

5.

1. 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.
2. Use (a) to find solutions \(y_1\), \(y_2\), and \(y_3\) of (A) such that \[\begin{array}{rl} y_1(1)&=1,\quad y_1'(1)=0,\quad y_1''(1)=0 \\[4pt][5 pt] y_2(1)&=0,\quad y_2'(1)=1,\quad y_2''(1)=0 \\[4pt][5 pt] y_3(1)&=0,\quad y_3'(1)=0,\quad y_3''(1)=1. \end{array}\nonumber \]
3. Use (b) to find the solution of (A) such that \[y(1)=k_0,\quad y'(1)=k_1,\quad y''(1)=k_2.\nonumber \]

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. \(y'''+y''-y'-y=0; \quad\{e^x,\,e^{-x},\,xe^{-x}\}\)
2. \(y'''-3y''+7y'-5y=0; \quad\{e^x,\,e^x\cos2x,\,e^x\sin2x\}\).
3. \(xy'''-y''-xy'+y=0; \quad \{e^x,\,e^{-x},\,x\}\)
4. \(x^2y'''+2xy''-(x^2+2)y=0; \quad \{e^x/ x,\,e^{-x}/ x,\,1\}\)
5. \((x^2-2x+2)y'''-x^2y''+2xy'-2y=0; \quad \{x,\,x^2,\,e^x\} \)
6. \((2x-1)y^{(4)}-4xy'''+(5-2x)y''+4xy'-4y=0; \quad\{x,\,e^x,\,e^{-x},e^{2x}\}\)
7. \(xy^{(4)}-y'''-4xy'+4y'=0; \quad\{1,x^2,\,e^{2x},\,e^{-2x}\}\)

7. Find the Wronskian \(W\) of a set of three solutions of \[y'''+2xy''+e^xy'-y=0,\nonumber \] given that \(W(0)=2\).

8. Find the Wronskian \(W\) of a set of four solutions of \[y^{(4)}+(\tan x)y'''+x^2y''+2xy=0,\nonumber \] given that \(W(\pi/4)=K\).

9.

1. Evaluate the Wronskian \(W\) \(\{e^x,\,xe^x,\, x^2e^x\}\). Evaluate \(W(0)\).
2. Verify that \(y_1\), \(y_2\), and \(y_3\) satisfy \[y'''-3y''+3y'-y=0. \tag{A} \]
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. \(\{1,\,e^x,\,e^{-x}\}\)
2. \(\{e^x,\, e^x\sin x,\,e^x\cos x\}\)
3. \(\{2,\,x+1,\,x^2+2\}\)
4. \(x,\,x\ln x,\,1/x\}\)
5. \(\{1,\,x,\,{x^2\over2!},\, {x^3\over3!}\,,\cdots,\,{x^n\over n!}\}\)
6. \(\{e^x,\,e^{-x},\,x\}\)
7. \(\{e^x/x,\,e^{-x}/x,\,1\}\)
8. \(\{x,\,x^2,\,e^x\}\)
9. \(\{x,\,x^3,\,1/x,\,1/x^2\}\)
10. \(\{e^x,\,e^{-x},\,x,\,e^{2x}\}\)
11. \(\{e^{2x},\,e^{-2x},\,1,\,x^2\}\)

11. Suppose \(Ly=0\) is normal on \((a,b)\) and \(x_0\) is in \((a,b)\). Use Theorem 9.1.1 to show 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 \((a,b)\).

12. Prove: If \(y_1\), \(y_2\), …, \(y_n\) are solutions of \(Ly=0\) and the functions \[z_i=\sum^n_{j=1}a_{ij}y_j,\quad 1\le i\le n,\nonumber \] form a fundamental set of solutions of \(Ly=0\), then so do \(y_1\), \(y_2\), …, \(y_n\).

13. Prove: If \[y=c_1y_1+c_2y_2+\cdots+c_ky_k+y_p\nonumber \] is a solution of a linear equation \(Ly=F\) for every choice of the constants \(c_1\), \(c_2\),…, \(c_k\), then \(Ly_i=0\) for \(1\le i\le k\).

14. Suppose \(Ly=0\) is normal on \((a,b)\) and let \(x_0\) be in \((a,b)\). For \(1\le i\le n\), let \(y_i\) be the solution of the initial value problem \[Ly_i=0, \quad y_i^{(j)} (x_0)= \left\{\begin{array}{cl} 0,& j\ne i-1,\\[4pt] 1,&j=i-1,\end{array}\right. 1\le i\le n,\nonumber \] where \(x_0\) is an arbitrary point in \((a,b)\). Show that any solution of \(Ly=0\) on \((a, b)\), can be written as \[y=c_1y_1+c_2y_2+\cdots+c_ny_n,\nonumber \] with \(c_j=y^{(j-1)}(x_0)\).

15. Suppose \(\{y_1, y_2,\dots, y_n\}\) is a fundamental set of solutions of \[\ P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=0\nonumber \] on \((a,b)\), and let \[\begin{array}{rl} z_1&=a_{11}y_1+a_{12}y_2+\cdots+a_{1n}y_n\\[4pt] z_2&=a_{21}y_1+a_{22}y_2+\cdots+a_{2n}y_n\\[4pt] \phantom{z_1}&\vdots\phantom{_1y_1+a}\vdots \phantom{_2y_2+\cdots+a}\vdots\phantom{_ny_n} \phantom{=b}\vdots\\[4pt] z_n&=a_{n1}y_1+a_{n2}y_2+\cdots+a_{nn}y_n, \end{array}\nonumber \] where the \(\{a_{ij}\}\) are constants. Show that \(\{z_1, z_2,\dots, z_n\}\) is a fundamental set of solutions of (A) if and only if the determinant \[\left|\begin{array}{cccc} a_{11}&a_{12}&\cdots&a_{1n}\\[4pt] a_{21}&a_{22}&\cdots&a_{2n}\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right|\nonumber \] is nonzero. HINT: The determinant of a product of \(n\times n\) matrices equals the product of the determinants.

16. Show that \(\{y_1,y_2,\dots,y_n\}\) is linearly dependent on \((a,b)\) if and only if at least one of the functions \(y_1\), \(y_2\), …, \(y_n\) 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, \[\left|\begin{array}{cccc}{a_{11}}&{a_{12}}&{\cdots }&{a_{1n}}\\[4pt]{a_{21}}&{a_{22}}&{\cdots }&{a_{2n}}\\[4pt]{\vdots }&{\vdots }&{\ddots }&{\vdots }\\[4pt]{a_{n1}}&{a_{n2}}&{\cdots }&{a_{nn}} \end{array} \right| = \sum\pm a_{1i_{1}}a_{2i_{2}},\cdots , a_{ni_{n}},\nonumber \] where the sum is over all permutations \((i_{i}, i_{2}, \cdots , i_{n})\) of \((1,2,\cdots ,n)\) and the choice of \(+\) or \(-\) in each term depends only on the permutation associated with that term.

17. Prove: If \[A(u_1,u_2,\dots,u_n)= \left|\begin{array}{cccc} a_{11}&a_{12}&\cdots&a_{1n}\\[4pt] a_{21}&a_{22}&\cdots&a_{2n}\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] a_{n-1,1}&a_{n-1,2}&\cdots&a_{n-1,n}\\[4pt] u_1&u_2&\cdots&u_n\end{array}\right|,\nonumber \] then \[A(u_1+v_1, u_2+v_2,\dots, u_n+v_n)=A(u_1,u_2,\dots,u_n)+A(v_1,v_2,\dots, v_n).\nonumber \]

18. Let \[F=\left|\begin{array}{cccc} f_{11}&f_{12}&\cdots&f_{1n}\\[4pt] f_{21}&f_{22}&\cdots&f_{2n}\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] f_{n1}&f_{n2}&\cdots&f_{nn}\end{array}\right|,\nonumber \] where \(f_{ij}\; (1\le i,\; j\le n)\) is differentiable. Show that \[F'=F_1+F_2+\cdots+F_n,\nonumber \] where \(F_i\) is the determinant obtained by differentiating the \(i\)th row of \(F\).

19. Use Exercise 9.1.18 to show that if \(W\) is the Wronskian of the \(n\)-times differentiable functions \(y_1\), \(y_2\), …, \(y_n\), then

\[W'= \left|\begin{array}{cccc} y_1&y_2&\cdots&y_n\\[4pt] y'_1&y'_2&\cdots&y'_n\\[4pt] \vdots&\vdots&\ddots&\vdots\\[4pt] y_1^{(n-2)}&y_2^{(n-2)}&\cdots&y_n^{(n-2)}\\[4pt] y_1^{(n)}&y_2^{(n)}&\cdots&y_n^{(n)} \end{array}\right|.\nonumber \]

Q9.1.3

20. Use Exercises 9.1.17 and 9.1.19 to show that if \(W\) is the Wronskian of solutions \(\{y_1,y_2,\dots,y_n\}\) of the normal equation \[P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+\cdots+P_n(x)y=0, \tag{A} \] then \(W'=-P_1W/P_0\). 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 \(\{y_1,y_2,\dots,y_n\}\) has no zeros in \((a,b)\), then the differential equation obtained by expanding the determinant \[\left|\begin{array}{ccccc} y&y_1&y_2&\cdots&y_n\\[4pt] y'&y'_1&y'_2&\cdots&y'_n\\[4pt] \vdots&\vdots&\vdots&\ddots& \vdots\\[4pt] y^{(n)}&y_{1}^{(n)}&y_2^{(n)}&\cdots&y_n^{(n)} \end{array}\right|=0,\nonumber \] in cofactors of its first column is normal and has \(\{y_1,y_2,\dots,y_n\}\) 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.

1. \(\{x,\,x^2-1,\,x^2+1\}\)
2. \(\{e^x,\,e^{-x},\,x\}\)
3. \(\{e^x,\,xe^{-x},\,1\}\)
4. \(\{x,\,x^2,\,e^x\}\)
5. \(\{x,\,x^2,\,1/x\}\)
6. \(\{x+1,\,e^x,\,e^{3x}\}\)
7. \(\{x,\,x^3,\,1/x,\,1/x^2\}\)
8. \(\{x,\,x\ln x,\,1/x,\,x^2\}\)
9. \(\{e^x,\,e^{-x},\,x,\,e^{2x}\}\)
10. \(\{e^{2x},\,e^{-2x},\,1,\,x^2\}\)