2.5.2 Problem 3

Internal problem ID [18481]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 3
Date solved : Thursday, March 13, 2025 at 12:06:22 PM
CAS classification : [_quadrature]

Solve

\begin{align*} 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}

Solved as first order ode of type dAlembert

Time used: 0.070 (sec)

Let \(p=y^{\prime }\) the ode becomes

\begin{align*} 2 p^{3}+p^{2}-y = 0 \end{align*}

Solving for \(y\) from the above results in

\begin{align*} \tag{1} y &= 2 p^{3}+p^{2} \\ \end{align*}

This has the form

\begin{align*} y=xf(p)+g(p)\tag {*} \end{align*}

Where \(f,g\) are functions of \(p=y'(x)\). The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. \(x\) gives

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(y=x f + g\) to (1A) shows that

\begin{align*} f &= 0\\ g &= 2 p^{3}+p^{2} \end{align*}

Hence (2) becomes

\begin{equation} \tag{2A} p = \left (6 p^{2}+2 p \right ) p^{\prime }\left (x \right ) \end{equation}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=0 \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = 0 \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{equation} \tag{3} p^{\prime }\left (x \right ) = \frac {p \left (x \right )}{6 p \left (x \right )^{2}+2 p \left (x \right )} \end{equation}

This ODE is now solved for \(p \left (x \right )\). No inversion is needed.

Integrating gives

\begin{align*} \int \left (6 p +2\right )d p &= dx\\ 3 p^{2}+2 p&= x +c_1 \end{align*}

Substituing the above solution for \(p\) in (2A) gives

\begin{align*} y &= 2 \left (-\frac {1}{3}+\frac {\sqrt {1+3 c_1 +3 x}}{3}\right )^{3}+\left (-\frac {1}{3}+\frac {\sqrt {1+3 c_1 +3 x}}{3}\right )^{2} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= 0 \\ y &= 2 \left (-\frac {1}{3}+\frac {\sqrt {1+3 c_1 +3 x}}{3}\right )^{3}+\left (-\frac {1}{3}+\frac {\sqrt {1+3 c_1 +3 x}}{3}\right )^{2} \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 385

ode:=2*diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= 0 \\ -6 \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}}-\sqrt {3}\, \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {-72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \sqrt {3}+3 i x -3 i c_{1}}{\sqrt {3}+3 i} &= 0 \\ \frac {72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (-i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-x +c_{1} \right ) \sqrt {3}+3 i x -3 i c_{1}}{-\sqrt {3}+3 i} &= 0 \\ \end{align*}

Maple trace

`Methods for first order ODEs: 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   -> Solving 1st order ODE of high degree, 1st attempt 
   trying 1st order WeierstrassP solution for high degree ODE 
   trying 1st order WeierstrassPPrime solution for high degree ODE 
   trying 1st order JacobiSN solution for high degree ODE 
   trying 1st order ODE linearizable_by_differentiation 
   trying differential order: 1; missing variables 
   <- differential order: 1; missing  x  successful`
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y=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} \\ {} & {} & \left [y^{\prime }=\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}, y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\textit {\_C1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =x +\mathit {C1} , \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\mathit {C1} , \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\mathit {C1} \right \} \end {array} \]

✗ Mathematica

ode=2*D[y[x],x]^3+D[y[x],x]^2-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✓ Sympy. Time used: 63.755 (sec). Leaf size: 343

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + 2*Derivative(y(x), x)**3 + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 2 i\right )}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 2 i\right )}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1\right )^{\frac {2}{3}} + \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 1}\, dy = C_{1} - \frac {x}{6}\right ] \]