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. $\{e^{2x},\,e^{-2x},\,1,\,x^2\}$