ODE
\[ 4 x y'(x)^2+4 y(x) y'(x)=1 \] ODE Classification
[[_homogeneous, `class G`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.494721 (sec), leaf count = 2029
\[\left \{\left \{y(x)\to -\frac {\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x+\sqrt {72 x+\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}-\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {2 e^{3 c_1}}{x^2}+\frac {2 e^{\frac {3 c_1}{2}} \left (e^{3 c_1}-432 x^3\right )}{\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x^3}} x+e^{\frac {3 c_1}{2}}}{18 x}\right \},\left \{y(x)\to -\frac {\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x-\sqrt {72 x+\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}-\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {2 e^{3 c_1}}{x^2}+\frac {2 e^{\frac {3 c_1}{2}} \left (e^{3 c_1}-432 x^3\right )}{\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x^3}} x+e^{\frac {3 c_1}{2}}}{18 x}\right \},\left \{y(x)\to -\frac {-\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x+\sqrt {72 x+\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}-\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {2 e^{3 c_1}}{x^2}-\frac {2 e^{\frac {3 c_1}{2}} \left (e^{3 c_1}-432 x^3\right )}{\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x^3}} x+e^{\frac {3 c_1}{2}}}{18 x}\right \},\left \{y(x)\to \frac {\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x+\sqrt {72 x+\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}-\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {2 e^{3 c_1}}{x^2}-\frac {2 e^{\frac {3 c_1}{2}} \left (e^{3 c_1}-432 x^3\right )}{\sqrt {36 x-\frac {36\ 2^{2/3} \left (e^{3 c_1}-2 x^3\right )}{\sqrt [3]{32 x^6+40 e^{3 c_1} x^3-e^{6 c_1}+\sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}+\frac {9 \sqrt [3]{64 x^6+80 e^{3 c_1} x^3-2 e^{6 c_1}+2 \sqrt {e^{3 c_1} \left (16 x^3+e^{3 c_1}\right ){}^3}}}{x}+\frac {e^{3 c_1}}{x^2}} x^3}} x-e^{\frac {3 c_1}{2}}}{18 x}\right \}\right \}\]
Maple ✓
cpu = 0.127 (sec), leaf count = 126
\[\left [-\frac {\textit {\_C1} \left (\frac {-y \left (x \right )+\sqrt {x +y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}} x^{2}}{\left (-y \left (x \right )+\sqrt {x +y \left (x \right )^{2}}\right )^{2}}+x -\frac {x^{2}}{3 \left (-y \left (x \right )+\sqrt {x +y \left (x \right )^{2}}\right )^{2}} = 0, \frac {\left (\frac {-2 y \left (x \right )-2 \sqrt {x +y \left (x \right )^{2}}}{x}\right )^{\frac {3}{2}} x^{2} \textit {\_C1}}{\left (y \left (x \right )+\sqrt {x +y \left (x \right )^{2}}\right )^{2}}+x -\frac {x^{2}}{3 \left (y \left (x \right )+\sqrt {x +y \left (x \right )^{2}}\right )^{2}} = 0\right ]\] Mathematica raw input
DSolve[4*y[x]*y'[x] + 4*x*y'[x]^2 == 1,y[x],x]
Mathematica raw output
{{y[x] -> -1/18*(E^((3*C[1])/2) + x*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x] + x*Sqrt[(2*E^(3*C[1]))/x^2 + 72*x + (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) - (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x + (2*E^((3*C[1])/2)*(E^(3*C[1]) - 432*x^3))/(x^3*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x])])/x}, {y[x] -> -1/18*(E^((3*C[1])/2) + x*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x] - x*Sqrt[(2*E^(3*C[1]))/x^2 + 72*x + (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) - (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x + (2*E^((3*C[1])/2)*(E^(3*C[1]) - 432*x^3))/(x^3*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x])])/x}, {y[x] -> -1/18*(E^((3*C[1])/2) - x*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x] + x*Sqrt[(2*E^(3*C[1]))/x^2 + 72*x + (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) - (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x - (2*E^((3*C[1])/2)*(E^(3*C[1]) - 432*x^3))/(x^3*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x])])/x}, {y[x] -> (-E^((3*C[1])/2) + x*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x] + x*Sqrt[(2*E^(3*C[1]))/x^2 + 72*x + (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) - (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x - (2*E^((3*C[1])/2)*(E^(3*C[1]) - 432*x^3))/(x^3*Sqrt[E^(3*C[1])/x^2 + 36*x - (36*2^(2/3)*(E^(3*C[1]) - 2*x^3))/(-E^(6*C[1]) + 40*E^(3*C[1])*x^3 + 32*x^6 + Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3) + (9*(-2*E^(6*C[1]) + 80*E^(3*C[1])*x^3 + 64*x^6 + 2*Sqrt[E^(3*C[1])*(E^(3*C[1]) + 16*x^3)^3])^(1/3))/x])])/(18*x)}}
Maple raw input
dsolve(4*x*diff(y(x),x)^2+4*y(x)*diff(y(x),x) = 1, y(x))
Maple raw output
[-_C1*((-y(x)+(x+y(x)^2)^(1/2))/x)^(3/2)/(-y(x)+(x+y(x)^2)^(1/2))^2*x^2+x-1/3/(-y(x)+(x+y(x)^2)^(1/2))^2*x^2 = 0, (1/x*(-2*y(x)-2*(x+y(x)^2)^(1/2)))^(3/2)/(y(x)+(x+y(x)^2)^(1/2))^2*x^2*_C1+x-1/3/(y(x)+(x+y(x)^2)^(1/2))^2*x^2 = 0]