Internal problem ID [2612]
Internal file name [OUTPUT/2104_Sunday_June_05_2022_02_48_44_AM_47541874/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.2, Basic Ideas and Terminology.
page 21
Problem number: Problem 34.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }=\frac {1}{x^{\frac {2}{3}}}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{x^{\frac {2}{3}}}\,\mathop {\mathrm {d}x}}\\ &= 3 x^{\frac {1}{3}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 x^{\frac {1}{3}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = 3 x^{\frac {1}{3}}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{x^{\frac {2}{3}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{x^{\frac {2}{3}}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=3 x^{\frac {1}{3}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=3 x^{\frac {1}{3}}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 11
dsolve(diff(y(x),x)=x^(-2/3),y(x), singsol=all)
\[ y \left (x \right ) = 3 x^{\frac {1}{3}}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 15
DSolve[y'[x]==x^(-2/3),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 3 \sqrt [3]{x}+c_1 \]