82.18.11 problem Ex. 12

Internal problem ID [18833]
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 : Tuesday, January 28, 2025 at 12:24:07 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 ({y^{\prime }}^{2}+1\right ) \end{align*}

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 106

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

\begin{align*} y \left (x \right ) &= \sqrt {-n}\, x \\ y \left (x \right ) &= -\sqrt {-n}\, x \\ y \left (x \right ) &= \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 \left (x \right ) &= \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*}

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 113

DSolve[(D[y[x],x]*y[x]+n*x)^2==(y[x]^2+n*x^2)*(1+ D[y[x],x]^2 ),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*}