29.33.23 problem 986

Internal problem ID [5562]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 986
Date solved : Monday, January 27, 2025 at 11:54:53 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+x^{2}+\left (-4 a^{2}+1\right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.213 (sec). Leaf size: 136

dsolve(((-4*a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2-8*a^2*x*y(x)*diff(y(x),x)+x^2+(-4*a^2+1)*y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}-\sqrt {\left (4 a^{2}-1\right ) \left (\textit {\_a}^{2}+1\right )^{2}}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}+\sqrt {\left (4 a^{2}-1\right ) \left (\textit {\_a}^{2}+1\right )^{2}}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} +c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.551 (sec). Leaf size: 328

DSolve[((1-4 a^2)x^2+y[x]^2) (D[y[x],x])^2 - 8 a^2 x y[x] D[y[x],x]+x^2+(1-4 a^2)y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

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