1.31 problem 29 part(a)

Internal problem ID [4942]
Internal file name [OUTPUT/4435_Sunday_June_05_2022_02_56_36_PM_45107296/index.tex]

Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section: Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number: 29 part(a).
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]

\[ \boxed {y^{\prime }-y^{\frac {1}{3}}=0} \]

1.31.1 Solving as quadrature ode

Integrating both sides gives \begin{align*} \int \frac {1}{y^{\frac {1}{3}}}d y &= \int d x \\ \frac {3 y^{\frac {2}{3}}}{2}&=x +c_{1} \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {3 y^{\frac {2}{3}}}{2} &= x +c_{1} \\ \end{align*}

Figure 42: Slope field plot

Verification of solutions

\[ \frac {3 y^{\frac {2}{3}}}{2} = x +c_{1} \] Verified OK.

1.31.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{\frac {1}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{\frac {1}{3}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{\frac {1}{3}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{\frac {1}{3}}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {3 y^{\frac {2}{3}}}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\left (6 x +6 c_{1} \right )^{\frac {3}{2}}}{27} \end {array} \]

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)=y(x)^(1/3),y(x), singsol=all)
 

\[ y \left (x \right )^{\frac {2}{3}}-\frac {2 x}{3}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.165 (sec). Leaf size: 29

DSolve[y'[x]==y[x]^(1/3),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2}{3} \sqrt {\frac {2}{3}} (x+c_1){}^{3/2} \\ y(x)\to 0 \\ \end{align*}