ODE
\[ (x-y(x)) \sqrt {y'(x)}=a \left (y'(x)+1\right ) \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 5.87877 (sec), leaf count = 611
\[\left \{\left \{y(x)\to -\frac {a^2}{x+c_1}-c_1\right \},\text {Solve}\left [c_1=\int _1^x\frac {2 a^2 \left (2 a^2-K[1]^2-y(x)^2+2 K[1] y(x)+\sqrt {(K[1]-y(x))^4-4 a^2 (K[1]-y(x))^2}\right ) \sqrt {-\frac {-4 a^2+K[1]^2+y(x)^2-2 K[1] y(x)}{a^2}} \left (K[1]-y(x)+\sqrt {-4 a^2+K[1]^2+y(x)^2-2 K[1] y(x)}\right )^{\frac {\sqrt {(K[1]-y(x))^4-4 a^2 (K[1]-y(x))^2}}{(K[1]-y(x)) \sqrt {-4 a^2+K[1]^2+y(x)^2-2 K[1] y(x)}}}}{(2 a+K[1]-y(x)) (2 a-K[1]+y(x))}dK[1]+\int _1^{y(x)}\left (\frac {4 a^4 \sqrt {\frac {4 a^2-(x-K[2])^2}{a^2}} \left (x-K[2]+\sqrt {(x-K[2])^2-4 a^2}\right )^{\frac {\sqrt {(x-K[2])^4-4 a^2 (x-K[2])^2}}{\sqrt {(x-K[2])^2-4 a^2} (x-K[2])}}}{(2 a+x-K[2]) (2 a-x+K[2])}-\int _1^x\frac {4 a^2 (K[1]-K[2]) \left (K[1]-K[2]+\sqrt {(K[1]-K[2])^2-4 a^2}\right )^{\frac {\sqrt {(K[1]-K[2])^4-4 a^2 (K[1]-K[2])^2}}{\sqrt {(K[1]-K[2])^2-4 a^2} (K[1]-K[2])}} \left (4 a^2-K[1]^2-K[2]^2+2 K[1] K[2]+\sqrt {(K[1]-K[2])^4-4 a^2 (K[1]-K[2])^2}\right )}{\sqrt {\frac {4 a^2-(K[1]-K[2])^2}{a^2}} \sqrt {(K[1]-K[2])^4-4 a^2 (K[1]-K[2])^2} (2 a+K[1]-K[2]) (2 a-K[1]+K[2])}dK[1]\right )dK[2],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.914 (sec), leaf count = 44
\[\left [y \left (x \right ) = x -2 a, y \left (x \right ) = x +\frac {-\frac {a^{3}}{\left (\textit {\_C1} -x \right )^{2}}-a}{\sqrt {\frac {a^{2}}{\left (\textit {\_C1} -x \right )^{2}}}}\right ]\] Mathematica raw input
DSolve[(x - y[x])*Sqrt[y'[x]] == a*(1 + y'[x]),y[x],x]
Mathematica raw output
{{y[x] -> -C[1] - a^2/(x + C[1])}, Solve[C[1] == Inactive[Integrate][(2*a^2*(2*a^2 - K[1]^2 + Sqrt[-4*a^2*(K[1] - y[x])^2 + (K[1] - y[x])^4] + 2*K[1]*y[x] - y[x]^2)*Sqrt[-((-4*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2)/a^2)]*(K[1] - y[x] + Sqrt[-4*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2])^(Sqrt[-4*a^2*(K[1] - y[x])^2 + (K[1] - y[x])^4]/((K[1] - y[x])*Sqrt[-4*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2])))/((2*a + K[1] - y[x])*(2*a - K[1] + y[x])), {K[1], 1, x}] + Inactive[Integrate][(4*a^4*Sqrt[(4*a^2 - (x - K[2])^2)/a^2]*(x + Sqrt[-4*a^2 + (x - K[2])^2] - K[2])^(Sqrt[-4*a^2*(x - K[2])^2 + (x - K[2])^4]/(Sqrt[-4*a^2 + (x - K[2])^2]*(x - K[2]))))/((2*a + x - K[2])*(2*a - x + K[2])) - Inactive[Integrate][(4*a^2*(K[1] - K[2])*(K[1] + Sqrt[-4*a^2 + (K[1] - K[2])^2] - K[2])^(Sqrt[-4*a^2*(K[1] - K[2])^2 + (K[1] - K[2])^4]/(Sqrt[-4*a^2 + (K[1] - K[2])^2]*(K[1] - K[2])))*(4*a^2 - K[1]^2 + Sqrt[-4*a^2*(K[1] - K[2])^2 + (K[1] - K[2])^4] + 2*K[1]*K[2] - K[2]^2))/(Sqrt[(4*a^2 - (K[1] - K[2])^2)/a^2]*Sqrt[-4*a^2*(K[1] - K[2])^2 + (K[1] - K[2])^4]*(2*a + K[1] - K[2])*(2*a - K[1] + K[2])), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]]}
Maple raw input
dsolve((x-y(x))*diff(y(x),x)^(1/2) = a*(1+diff(y(x),x)), y(x))
Maple raw output
[y(x) = x-2*a, y(x) = x+(-a^3/(_C1-x)^2-a)/(a^2/(_C1-x)^2)^(1/2)]