29.34.2 problem 997

Internal problem ID [5573]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 997
Date solved : Monday, January 27, 2025 at 12:07:17 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.118 (sec). Leaf size: 198

dsolve((-a^2+1)*y(x)^2*diff(y(x),x)^2-3*a^2*x*y(x)*diff(y(x),x)-a^2*x^2+y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}-\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.180 (sec). Leaf size: 341

DSolve[(1-a^2)y[x]^2 (D[y[x],x])^2 -2 a^2 x y[x] D[y[x],x]-a^2 x y[x] D[y[x],x]-a^2 x^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {\log \left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )-\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}-\frac {2 y(x)^2}{x^2}\right )-\frac {2 \text {arctanh}\left (\frac {\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}-1}{\sqrt {5 a^4-2 a^2+1}}\right )}{\sqrt {5 a^4-2 a^2+1}}}{4 a^2-4}&=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}+1}{\sqrt {5 a^4-2 a^2+1}}\right )}{\sqrt {5 a^4-2 a^2+1}}+\log \left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )+\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}-\frac {2 y(x)^2}{x^2}\right )}{4 a^2-4}&=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ] \\ \end{align*}