Internal problem ID [8788]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 453.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (a^{2}-1\right ) x^{2} {y^{\prime }}^{2}+2 x y y^{\prime }-y^{2}=-a^{2} x^{2}} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 229
dsolve((a^2-1)*x^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)-y(x)^2+a^2*x^2 = 0,y(x), singsol=all)
\begin{align*} \frac {2 a \ln \left (x \right )-2 \sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {y \left (x \right )^{2}+x^{2}}{x^{2}}\right ) a -2 a c_{1} +2 \ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{2 a} &= 0 \\ \frac {2 a \ln \left (x \right )+2 \sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {y \left (x \right )^{2}+x^{2}}{x^{2}}\right ) a -2 a c_{1} -2 \ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{2 a} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.001 (sec). Leaf size: 223
DSolve[a^2*x^2 - y[x]^2 + 2*x*y[x]*y'[x] + (-1 + a^2)*x^2*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\ \end{align*}