82.18.11 problem Ex. 12

Internal problem ID [18762]
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. Examples on chapter III. page 38
Problem number : Ex. 12
Date solved : Monday, March 31, 2025 at 06:09:11 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (y^{\prime } y+n x \right )^{2}&=\left (y^{2}+n \,x^{2}\right ) \left (1+{y^{\prime }}^{2}\right ) \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 106

ode:=(y(x)*diff(y(x),x)+n*x)^2 = (y(x)^2+n*x^2)*(1+diff(y(x),x)^2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-n}\, x \\ y &= -\sqrt {-n}\, x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (n -1\right ) \left (\textit {\_a}^{2}+n \right ) n}}{\left (n -1\right ) \left (\textit {\_a}^{2}+n \right )}d \textit {\_a} +c_1 \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (n -1\right ) \left (\textit {\_a}^{2}+n \right ) n}}{\left (n -1\right ) \left (\textit {\_a}^{2}+n \right )}d \textit {\_a} +c_1 \right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.32 (sec). Leaf size: 113

ode=(D[y[x],x]*y[x]+n*x)^2==(y[x]^2+n*x^2)*(1+ D[y[x],x]^2 ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {n-1}{n}}+1}-\frac {1}{2} e^{-c_1} n x^{1-\sqrt {\frac {n-1}{n}}} \\ y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {n-1}{n}}} \left (-n x^{2 \sqrt {\frac {n-1}{n}}}+e^{2 c_1}\right ) \\ \end{align*}

✓ Sympy. Time used: 19.142 (sec). Leaf size: 673

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq((n*x + y(x)*Derivative(y(x), x))**2 - (n*x**2 + y(x)**2)*(Derivative(y(x), x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]