ODE
\[ \sqrt {\left (a x^2+y(x)^2\right ) \left (y'(x)^2+1\right )}-a x-y(x) y'(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.643731 (sec), leaf count = 241
\[\left \{\left \{y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (x^{2 \sqrt {\frac {a-1}{a}}}-e^{2 c_1}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right )\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \}\right \}\]
Maple ✓
cpu = 3.399 (sec), leaf count = 159
\[\left [y \left (x \right ) = -\frac {x^{-\frac {-a +\sqrt {a \left (a -1\right )}}{a}} a^{3}-x^{-\frac {-a +\sqrt {a \left (a -1\right )}}{a}} a^{2}-x^{\frac {\sqrt {a \left (a -1\right )}+a}{a}} \textit {\_C1}^{2}}{2 \sqrt {a \left (a -1\right )}\, \textit {\_C1}}, y \left (x \right ) = \frac {-x^{\frac {\sqrt {a \left (a -1\right )}+a}{a}} a^{3}+x^{-\frac {-a +\sqrt {a \left (a -1\right )}}{a}} \textit {\_C1}^{2}+x^{\frac {\sqrt {a \left (a -1\right )}+a}{a}} a^{2}}{2 \sqrt {a \left (a -1\right )}\, \textit {\_C1}}\right ]\] Mathematica raw input
DSolve[-(a*x) - y[x]*y'[x] + Sqrt[(a*x^2 + y[x]^2)*(1 + y'[x]^2)] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[a]*x^(1 - Sqrt[(-1 + a)/a])*(-E^(2*C[1]) + x^(2*Sqrt[(-1 + a)/a])))/(2*E^C[1])}, {y[x] -> (Sqrt[a]*x^(1 - Sqrt[(-1 + a)/a])*(E^(2*C[1]) - x^(2*Sqrt[(-1 + a)/a])))/(2*E^C[1])}, {y[x] -> -1/2*(Sqrt[a]*x^(1 - Sqrt[(-1 + a)/a])*(-1 + E^(2*C[1])*x^(2*Sqrt[(-1 + a)/a])))/E^C[1]}, {y[x] -> (Sqrt[a]*x^(1 - Sqrt[(-1 + a)/a])*(-1 + E^(2*C[1])*x^(2*Sqrt[(-1 + a)/a])))/(2*E^C[1])}}
Maple raw input
dsolve(((a*x^2+y(x)^2)*(1+diff(y(x),x)^2))^(1/2)-y(x)*diff(y(x),x)-a*x = 0, y(x))
Maple raw output
[y(x) = -1/2*(x^(-(-a+(a*(a-1))^(1/2))/a)*a^3-x^(-(-a+(a*(a-1))^(1/2))/a)*a^2-x^(((a*(a-1))^(1/2)+a)/a)*_C1^2)/(a*(a-1))^(1/2)/_C1, y(x) = 1/2*(-x^(((a*(a-1))^(1/2)+a)/a)*a^3+x^(-(-a+(a*(a-1))^(1/2))/a)*_C1^2+x^(((a*(a-1))^(1/2)+a)/a)*a^2)/(a*(a-1))^(1/2)/_C1]