Internal problem ID [4218]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 986.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+\left (-4 a^{2}+1\right ) y^{2}=-x^{2}} \]
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 154
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 )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}+\sqrt {4 \textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+8 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+4 a^{2}-1}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right )+c_{1} \right ) x \end{align*}
✓ Solution by Mathematica
Time used: 1.499 (sec). Leaf size: 328
DSolve[((1-4 a^2)x^2+y[x]^2) (y'[x])^2 - 8 a^2 x y[x] y'[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*}