3.1E: Exercises

Last updated
Save as PDF

Page ID: 18789

$trench-1995.jpg$
Contributed by William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\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]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \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]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$

Exercises

Exercise $\PageIndex{1}$

Find the order of the equation.

(a) $\displaystyle{d^2y\over dx^2}+2{dy\over dx}\ {d^3y\over dx^3}+x=0$

(b) $y''-3y'+2y=x^7$

(d) $y''y-(y')^2=2$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{2}$

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.

(a) $y=ce^{2x}; \quad y'=2y$

(b) $y={x^2\over3}+{c\over x}; \quad xy'+y=x^2$

(d) $y=(1+ce^{-x^2/2}); (1-ce^{-x^2/2})^{-1} \quad 2y'+x(y^2-1)=0$

(e) $y={\tan\left( {x^3\over3}+c\right)}; \quad y'=x^2(1+y^2)$

(f) $y=(c_1+c_2x)e^x+\sin x+x^2; \quad y''-2y'+y=-2 \cos x+x^2-4x+2$

(g) $y=c_1e^x+c_2x+{2\over x}; \quad (1-x)y''+xy'- y=4(1-x-x^2)x^{-3}$

(h) $y=x^{-1/2}(c_1\sin x+c_2 \cos x)+4x+8$;

$x^2y''+xy'+{\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{3}$

Find all solutions of the equation.

(a) $y'=-x$

(b) $y'=-x \sin x$

(d) $y''=x \cos x$

(e) $y''=2xe^x$

(f) $y''=2x+\sin x+e^x$

(g) $y'''=-\cos x$

(h) $y'''=-x^2+e^x$

(i) $y'''=7e^{4x}$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{4}$

Solve the initial value problem.

(a) $y'=-xe^x, \quad y(0)=1$

(b) $y'=x \sin x^2, \quad y\left({\sqrt{\pi\over2}}\right)=1$

(d) $y''=x^4, \quad y(2)=-1, \quad y'(2)=-1$

(e) $y''=xe^{2x}, \quad y(0)=7, \quad y'(0)=1$

(f) $y''=- x \sin x, \quad y(0)=1, \quad y'(0)=-3$

(g) $y'''=x^2e^x, \quad y(0)=1, \quad y'(0)=-2, \quad y''(0)=3$

(h) $y'''=2+\sin 2x, \quad y(0)=1, \quad y'(0)=-6, \quad y''(0)=3$

(i) $y'''=2x+1, \quad y(2)=1, \quad y'(2)=-4, \quad y''(2)=7$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{5}$

Verify that the function is a solution of the initial value problem.

(a) $y=x\cos x; \quad y'=\cos x-y\tan x, \quad y(\pi/4)={\pi\over4\sqrt{2}}$

(b) ${y={1+2\ln x\over x^2}+{1\over2}; \quad y'={x^2-2x^2y+2\over x^3}, \quad y(1)={3\over2}}$

(d) ${y={2\over x-2}; \quad y'={-y(y+1)\over x}}, \quad y(1)=-2$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{6}$

Verify that the function is a solution of the initial value problem.

(a) $y=x^2(1+\ln x); \quad y''={3xy'-4y\over x^2}, \quad y(e)=2e^2, \quad y'(e)=5e$

(b) $y={x^2\over3}+x-1; \quad y''={x^2-xy'+y+1\over x^2}, \quad y(1)={1\over3}, \quad y'(1)={5\over3}$

(d) $y={x^2\over 1-x}; \quad y''={2(x+y)(xy'-y)\over x^3}, \quad y(1/2)=1/2, \quad y'(1/2)=3$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{7}$

Suppose an object is launched from a point 320 feet above the earth with an initial velocity of 128 ft/sec upward, and the only force acting on it thereafter is gravity. Take $g=32 ft/sec^2$

(a) Find the highest altitude attained by the object.

(b) Determine how long it takes for the object to fall to the ground.

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{8}$

Let $a$ be a nonzero real number.

(a) Verify that if $c$ is an arbitrary constant then equation A: $y=(x-c)^a $ is a solution of equation B: $y'=ay^{(a-1)/a}$ on $(c,\infty)$.

(b) Suppose $a<0$ or $a>1$. Can you think of a solution of (B) that isn't of the form (A)?

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{9}$

Verify that $y= e^x-1, x \ge 0$ and $1-e^{-x}, x < 0, $ is a solution of $y'=|y|+1$ on $(-\infty,\infty)$.

Hint: Use the definition of derivative at $x=0$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{10}$

(a) Verify that if $c$ is any real number then equation A: $y=c^2+cx+2c+1$ satisfies equation B: $y'={-(x+2)+\sqrt{x^2+4x+4y}\over2}$ on some open interval. Identify the open interval.

(b) Verify that $y_1={-x(x+4)\over4}$ also satisfies (B) on some open interval, and identify the open interval. (Note that $y_1$ can't be obtained by selecting a value of $c$ in (A).

Answer: Add texts here. Do not delete this text first

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)