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 = 0.185505 (sec), leaf count = 113
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (e^{2 c_1}-a x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \},\left \{y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {a-1}{a}}+1}-\frac {1}{2} a e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}}\right \}\right \}\]
Maple ✓
cpu = 0.384 (sec), leaf count = 176
\[ \left \{ {\frac {1}{x} \left ( \left ( \left ( {a}^{2}-a \right ) y \left ( x \right ) +\sqrt {a \left ( a-1 \right ) \left ( a{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }\sqrt {{a}^{2}-a} \right ) {x}^{{\frac {1}{a}\sqrt {a \left ( a-1 \right ) }}}-{\it \_C1}\,\sqrt {{a}^{2}-a}x \right ) {\frac {1}{\sqrt {{a}^{2}-a}}}}=0,{\frac {1}{x} \left ( -{\it \_C1}\,\sqrt {{a}^{2}-a}{x}^{{\frac {1}{a}\sqrt {a \left ( a-1 \right ) }}}x+ \left ( {a}^{2}-a \right ) y \left ( x \right ) +\sqrt {a \left ( a-1 \right ) \left ( a{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }\sqrt {{a}^{2}-a} \right ) {\frac {1}{\sqrt {{a}^{2}-a}}} \left ( {x}^{{\frac {1}{a}\sqrt {a \left ( a-1 \right ) }}} \right ) ^{-1}}=0 \right \} \] 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] -> (x^(1 - Sqrt[(-1 + a)/a])*(E^(2*C[1]) - a*x^(2*Sqrt[(-1 + a)/a])))/(2*E^C[1])}, {y[x] -> -(a*x^(1 - Sqrt[(-1 + a)/a]))/(2*E^C[1]) + (E^C[1]*x^(1 + Sqrt[(-1 + a)/a]))/2}}
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),'implicit')
Maple raw output
(((a^2-a)*y(x)+(a*(a-1)*(a*x^2+y(x)^2))^(1/2)*(a^2-a)^(1/2))*x^((a*(a-1))^(1/2)/a)-_C1*(a^2-a)^(1/2)*x)/(a^2-a)^(1/2)/x = 0, 1/(a^2-a)^(1/2)*(-_C1*(a^2-a)^(1/2)*x^((a*(a-1))^(1/2)/a)*x+(a^2-a)*y(x)+(a*(a-1)*(a*x^2+y(x)^2))^(1/2)*(a^2-a)^(1/2))/(x^((a*(a-1))^(1/2)/a))/x = 0