1.2E: Basic Concepts (Exercises)
- Page ID
- 134297
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Find the order of the equation.
a. \( {d^2y\over dx^2}+2 {dy\over dx}\ {d^3y\over dx^3}+x=0\)
b. \(y''-3y'+2y=x^7\)
c. \(y'-y^7=0\)
d. \(y''y-(y')^2=2\)
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. \(y= {x^2\over3} +{c\over x}; \quad xy'+y=x^2\)
b. \(y= {1\over2}+ce^{-x^2}; \quad y'+2xy=x\)
c. \(y= {\tan\left( {x^3\over3}+c\right)}; \quad y'=x^2(1+y^2)\)
d. \(y=x^{-1/2}(c_1\sin x+c_2 \cos x)+4x+8\); \(x^2y''+xy'+ {\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2\)
3. Find all solutions of the equation.
a. \(y'=-x\)
b. \(y''=x \cos x\)
c. \(y'''=7e^{4x}\)
4. Solve the initial value problem.
a. \(y'=-xe^x, \quad y(0)=1\)
b. \( {y'=x \sin(x^2), \quad y\left({\sqrt{\pi\over2}}\right)=1}\)
c. \(y'''=2+\sin(2x), \quad y(0)=1, \quad y'(0)=-6, \quad y''(0)=3\)
5. Verify that the function is a solution of the initial value problem.
a. \( {y={2\over x-2}; \quad y'={-y(y+1)\over x}}, \quad y(1)=-2\)
b. \(y= {x^2\over3}+x-1; \quad y''= {x^2-xy'+y+1\over x^2}, \quad y(1)= {1\over3}, \quad y'(1)= {5\over3}\)
6. 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\(^2\).
- Find the highest altitude attained by the object.
- Determine how long it takes for the object to fall to the ground.
7. Verify that \[\begin{aligned}y= \left\{ \begin{array}{cl} e^x-1,& x \ge 0, \\[6pt] 1-e^{-x},& x < 0, \end{array}\right.\end{aligned}\]
is a solution of
\[\begin{aligned}y'=|y|+1\end{aligned}\] on \((-\infty,\infty)\).