Internal problem ID [12601]
Internal file name [OUTPUT/11254_Thursday_October_19_2023_04_43_48_PM_68138707/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 1. Introduction. Exercises 1.3, page 27
Problem number: 6.
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]
\[ \boxed {y^{\prime }-3 y^{\frac {2}{3}}=0} \]
Integrating both sides gives \begin{align*} \int \frac {1}{3 y^{\frac {2}{3}}}d y &= \int d x \\ y^{\frac {1}{3}}&=x +c_{1} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y^{\frac {1}{3}} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y^{\frac {1}{3}} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-3 y^{\frac {2}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=3 y^{\frac {2}{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {2}{3}}}=3 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {2}{3}}}d x =\int 3d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 3 y^{\frac {1}{3}}=3 x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x^{3}+c_{1} x^{2}+\frac {1}{3} x \,c_{1}^{2}+\frac {1}{27} c_{1}^{3} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 14
dsolve(diff(y(x),x)=3*y(x)^(2/3),y(x), singsol=all)
\[ y \left (x \right )^{\frac {1}{3}}-c_{1} -x = 0 \]
✓ Solution by Mathematica
Time used: 0.166 (sec). Leaf size: 22
DSolve[y'[x]==3*y[x]^(2/3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{27} (3 x+c_1){}^3 \\ y(x)\to 0 \\ \end{align*}