82.14.1 problem Ex. 1

Internal problem ID [18736]
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 33
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:03:35 PM
CAS classification : [_dAlembert]

\begin{align*} x -y y^{\prime }&=a {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 396

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

\begin{align*} \frac {c_1 \left (y-\sqrt {4 a x +y^{2}}\right )}{\sqrt {\frac {-y+\sqrt {4 a x +y^{2}}+2 a}{a}}\, \sqrt {\frac {-y+\sqrt {4 a x +y^{2}}-2 a}{a}}}+x -\frac {\left (y-\sqrt {4 a x +y^{2}}\right ) \left (-3 \ln \left (2\right )+2 \ln \left (\frac {2 \sqrt {\frac {y^{2}-y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x}{a^{2}}}\, a -\sqrt {2}\, \left (y-\sqrt {4 a x +y^{2}}\right )}{a}\right )\right ) \sqrt {2}}{4 \sqrt {\frac {y^{2}-y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x}{a^{2}}}} &= 0 \\ \frac {c_1 \left (y+\sqrt {4 a x +y^{2}}\right )}{2 \sqrt {\frac {-y-\sqrt {4 a x +y^{2}}+2 a}{a}}\, \sqrt {\frac {-y-\sqrt {4 a x +y^{2}}-2 a}{a}}}+x -\frac {\left (y+\sqrt {4 a x +y^{2}}\right ) \sqrt {2}\, \left (-\frac {3 \ln \left (2\right )}{2}+\ln \left (\frac {2 \sqrt {\frac {y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x +y^{2}}{a^{2}}}\, a -\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right )}{a}\right )\right )}{2 \sqrt {\frac {y \sqrt {4 a x +y^{2}}-2 a^{2}+2 a x +y^{2}}{a^{2}}}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.775 (sec). Leaf size: 61

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

\[ \text {Solve}\left [\left \{x=\frac {a K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

Timed Out