$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

[exer:9.1.1] Verify that the given function is the solution of the initial value problem.

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

[exer:9.1.2] Solve the initial value problem

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

[exer:9.1.3] Solve the initial value problem

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

[exer:9.1.4] Find solutions $$y_1$$, $$y_2$$, …, $$y_n$$ of the equation $$y^{(n)}=0$$ that satisfy the initial conditions

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

[exer:9.1.5] 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

$\begin{array}{rl} y_1(1)&=1,\quad y_1'(1)=0,\quad y_1''(1)=0 \\[5 pt] y_2(1)&=0,\quad y_2'(1)=1,\quad y_2''(1)=0 \\[5 pt] y_3(1)&=0,\quad y_3'(1)=0,\quad y_3''(1)=1. \end{array}\nonumber$

Use (b) to find the solution of (A) such that $y(1)=k_0,\quad y'(1)=k_1,\quad y''(1)=k_2.\nonumber$

[exer:9.1.6] 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. $$xy^{(4)}-y'''-4xy'+4y'=0; \quad\{1,x^2,\,e^{2x},\,e^{-2x}\}$$

[exer:9.1.7] Find the Wronskian $$W$$ of a set of three solutions of $y'''+2xy''+e^xy'-y=0,\nonumber$ given that $$W(0)=2$$.

[exer:9.1.8] Find the Wronskian $$W$$ of a set of four solutions of $y^{(4)}+(\tan x)y'''+x^2y''+2xy=0,\nonumber$ given that $$W(\pi/4)=K$$.

[exer:9.1.9] Evaluate the Wronskian $$W$$ $$\{e^x,\,xe^x,\, x^2e^x\}$$. Evaluate $$W(0)$$. Verify that $$y_1$$, $$y_2$$, and $$y_3$$ satisfy $y'''-3y''+3y'-y=0. \tag{A}$

Use $$W(0)$$ from (a) and Abel’s formula to calculate $$W(x)$$.

What is the general solution of (A)?

[exer:9.1.10] 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. $$\{e^{2x},\,e^{-2x},\,1,\,x^2\}$$

[exer:9.1.11] Suppose $$Ly=0$$ is normal on $$(a,b)$$ and $$x_0$$ is in $$(a,b)$$. Use Theorem [thmtype: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 $$(a,b)$$.

[exer:9.1.12] 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 $$y_1$$, $$y_2$$, …, $$y_n$$.

[exer:9.1.13] 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 $$1\le i\le k$$.

[exer:9.1.14] 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 $Ly_i=0, \quad y_i^{(j)} (x_0)= \left\{\begin{array}{cl} 0,& j\ne i-1,\\ [5 pt] 1,&j=i-1,\end{array}\right. 1\le i\le n,\nonumber$ 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 $$c_j=y^{(j-1)}(x_0)$$.

[exer:9.1.15] 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 $\begin{array}{rl} z_1&=a_{11}y_1+a_{12}y_2+\cdots+a_{1n}y_n\\ z_2&=a_{21}y_1+a_{22}y_2+\cdots+a_{2n}y_n\\ \phantom{z_1}&\vdots\phantom{_1y_1+a}\vdots \phantom{_2y_2+\cdots+a}\vdots\phantom{_ny_n} \phantom{=b}\vdots\\ z_n&=a_{n1}y_1+a_{n2}y_2+\cdots+a_{nn}y_n, \end{array}\nonumber$ 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 $\left|\begin{array}{cccc} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right|\nonumber$ is nonzero.

[exer:9.1.16] 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 $$(a,b)$$.

[exer:9.1.17] 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 $A(u_1+v_1, u_2+v_2,\dots, u_n+v_n)=A(u_1,u_2,\dots,u_n)+A(v_1,v_2,\dots, v_n).\nonumber$

[exer:9.1.18] 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 $$F$$.

[exer:9.1.19] Use Exercise [exer:9.1.18} to show that if $$W$$ is the Wronskian of the $$n$$-times differentiable functions $$y_1$$, $$y_2$$, …, $$y_n$$, then

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

[exer:9.1.20] Use Exercises [exer:9.1.17} and [exer: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 (Eqn. Equation \ref{eq:9.1.15}) from this.

[exer:9.1.21] Prove Theorem [thmtype:9.1.6}.

[exer:9.1.22] Prove Theorem [thmtype:9.1.7}.

[exer:9.1.23] 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 $$(a,b)$$.

[exer:9.1.24] Use the method suggested by Exercise [exer: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. $$\{e^{2x},\,e^{-2x},\,1,\,x^2\}$$