4.2E: Introduction to Linear Higher Order Equations (Exercises)

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Jan 7, 2021
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- 4.2: Introduction to Linear Higher Order Equations
- 4.3: Constant Coefficient Homogeneous Equations

Zoya Kravets
Mission College

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Q4.2.1

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

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

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

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

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

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

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 $W(\pi/4)=K$ .

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)?

10. Compute the Wronskian of the given set of functions.

$\{1,\,e^x,\,e^{-x}\}$
$\{e^x,\, e^x\sin x,\,e^x\cos x\}$
$\{2,\,x+1,\,x^2+2\}$
$x,\,x\ln x,\,1/x\}$
$\{1,\,x,\,{x^2\over2!},\, {x^3\over3!}\,,\cdots,\,{x^n\over n!}\}$
$\{e^x,\,e^{-x},\,x\}$
$\{e^x/x,\,e^{-x}/x,\,1\}$
$\{x,\,x^2,\,e^x\}$
$\{x,\,x^3,\,1/x,\,1/x^2\}$
$\{e^x,\,e^{-x},\,x,\,e^{2x}\}$
$\{e^{2x},\,e^{-2x},\,1,\,x^2\}$

Q4.2.2

11. 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 $(a,b)$ .

12. Use the method suggested by Exercise 4.2.11 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.

$\{x,\,x^2-1,\,x^2+1\}$
$\{e^x,\,e^{-x},\,x\}$
$\{e^x,\,xe^{-x},\,1\}$
$\{x,\,x^2,\,e^x\}$
$\{x,\,x^2,\,1/x\}$
$\{x+1,\,e^x,\,e^{3x}\}$
$\{x,\,x^3,\,1/x,\,1/x^2\}$
$\{x,\,x\ln x,\,1/x,\,x^2\}$
$\{e^x,\,e^{-x},\,x,\,e^{2x}\}$
$\{e^{2x},\,e^{-2x},\,1,\,x^2\}$

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