76.15.1 problem Ex. 1

Internal problem ID [20096]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 35
Problem number : Ex. 1
Date solved : Thursday, October 02, 2025 at 05:25:42 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 106

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

\begin{align*} y &= \frac {\operatorname {LambertW}\left (-\sqrt {3}\, {\mathrm e}^{-1+\frac {x}{2}-\frac {c_1}{2}}\right ) \left (\operatorname {LambertW}\left (-\sqrt {3}\, {\mathrm e}^{-1+\frac {x}{2}-\frac {c_1}{2}}\right )+2\right )}{3} \\ y &= \frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-\textit {\_Z} -x +2 \,{\mathrm e}^{\textit {\_Z}}-2+c_1 -\ln \left (3\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}}{3}-\frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} -x +2 \,{\mathrm e}^{\textit {\_Z}}-2+c_1 -\ln \left (3\right )+\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}}{3} \\ \end{align*}

✓ Mathematica. Time used: 16.274 (sec). Leaf size: 86

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

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

✓ Sympy. Time used: 0.458 (sec). Leaf size: 58

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

\[ \left [ x + 2 \sqrt {3 y{\left (x \right )} + 1} - 2 \log {\left (\sqrt {3 y{\left (x \right )} + 1} + 1 \right )} = C_{1}, \ x - 2 \sqrt {3 y{\left (x \right )} + 1} - 2 \log {\left (\sqrt {3 y{\left (x \right )} + 1} - 1 \right )} = C_{1}\right ] \]