Internal problem ID [4161]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 926.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {\left (-a^{2}+1\right ) x^{2} {y^{\prime }}^{2}-2 y^{\prime } x y+y^{2}=a^{2} x^{2}} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 229
dsolve((-a^2+1)*x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-a^2*x^2+y(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 {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right ) a -2 c_{1} a +2 \ln \left (\frac {\sqrt {\frac {-x^{2} a^{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 {-x^{2} a^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x}\right )+\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right ) a -2 c_{1} a -2 \ln \left (\frac {\sqrt {\frac {-x^{2} a^{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: 1.04 (sec). Leaf size: 223
DSolve[(1-a^2)x^2 (y'[x])^2-2 x y[x] y'[x]-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*}