ODE
\[ (3 x+5) y'(x)^2-(3 y(x)+3) y'(x)+y(x)=0 \] ODE Classification
[_rational, _dAlembert]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 2.23676 (sec), leaf count = 70
\[\text {Solve}\left [\left \{3 e^{-3 K[1]} \left ((9 K[1]-3) \text {Ei}(3 K[1])+x e^{3 K[1]}-3 c_1 K[1]+c_1\right )=4,y(x)=\frac {K[1] ((3 x+5) K[1]-3)}{3 K[1]-1}\right \},\{y(x),K[1]\}\right ]\]
Maple ✓
cpu = 2.814 (sec), leaf count = 745
\[\left [\frac {\left (9 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}-18 x +27 y \left (x \right )-3\right ) {\mathrm e}^{-\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}} \textit {\_C1}}{5+3 x}+x -\frac {\left (27 \expIntegral \left (1, -\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}-54 \expIntegral \left (1, -\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) x +81 \expIntegral \left (1, -\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right ) y \left (x \right )+24 \,{\mathrm e}^{\frac {\frac {9 y \left (x \right )}{2}+\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}} x -9 \expIntegral \left (1, -\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}\right )+40 \,{\mathrm e}^{\frac {\frac {9 y \left (x \right )}{2}+\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}}\right ) {\mathrm e}^{-\frac {3 \left (3 y \left (x \right )+3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}}}{6 \left (5+3 x \right )} = 0, \frac {\left (9 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}+18 x -27 y \left (x \right )+3\right ) {\mathrm e}^{\frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}} \textit {\_C1}}{5+3 x}+x +\frac {\left (27 \expIntegral \left (1, \frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}+54 \expIntegral \left (1, \frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) x -81 \expIntegral \left (1, \frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right ) y \left (x \right )-24 \,{\mathrm e}^{-\frac {3 \left (-3 y \left (x \right )-3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}} x +9 \expIntegral \left (1, \frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}\right )-40 \,{\mathrm e}^{-\frac {3 \left (-3 y \left (x \right )-3+\sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}\right )}{2 \left (5+3 x \right )}}\right ) {\mathrm e}^{\frac {-\frac {9 y \left (x \right )}{2}-\frac {9}{2}+\frac {3 \sqrt {-12 x y \left (x \right )+9 y \left (x \right )^{2}-2 y \left (x \right )+9}}{2}}{5+3 x}}}{30+18 x} = 0\right ]\] Mathematica raw input
DSolve[y[x] - (3 + 3*y[x])*y'[x] + (5 + 3*x)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
Solve[{(3*(E^(3*K[1])*x + C[1] - 3*C[1]*K[1] + ExpIntegralEi[3*K[1]]*(-3 + 9*K[1])))/E^(3*K[1]) == 4, y[x] == (K[1]*(-3 + (5 + 3*x)*K[1]))/(-1 + 3*K[1])}, {y[x], K[1]}]
Maple raw input
dsolve((5+3*x)*diff(y(x),x)^2-(3+3*y(x))*diff(y(x),x)+y(x) = 0, y(x))
Maple raw output
[(9*(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)-18*x+27*y(x)-3)*exp(-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))/(5+3*x)*_C1+x-1/6*(27*Ei(1,-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))*(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)-54*Ei(1,-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))*x+81*Ei(1,-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))*y(x)+24*exp(3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))*x-9*Ei(1,-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))+40*exp(3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))))*exp(-3/2/(5+3*x)*(3*y(x)+3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)))/(5+3*x) = 0, (9*(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)+18*x-27*y(x)+3)*exp(3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))/(5+3*x)*_C1+x+1/6*(27*Ei(1,3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))*(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2)+54*Ei(1,3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))*x-81*Ei(1,3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))*y(x)-24*exp(-3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))*x+9*Ei(1,3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))-40*exp(-3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x)))*exp(3/2*(-3*y(x)-3+(-12*x*y(x)+9*y(x)^2-2*y(x)+9)^(1/2))/(5+3*x))/(5+3*x) = 0]