ODE
\[ a-b x+y'(x)^3+y'(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(x\)
Mathematica ✓
cpu = 11.2602 (sec), leaf count = 1862
\[\left \{\left \{y(x)\to \frac {-24 \sqrt [3]{2} 3^{2/3} \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^{2/3} \left (9 \sqrt {3} a-9 \sqrt {3} b x+\sqrt {27 (a-b x)^2+4}\right )-\frac {12 \sqrt {3} \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right ) \left (729 a^4-2916 b x a^3-81 \sqrt {3} \sqrt {27 (a-b x)^2+4} a^3+4374 b^2 x^2 a^2+243 \sqrt {3} b x \sqrt {27 (a-b x)^2+4} a^2+324 a^2-2916 b^3 x^3 a-648 b x a-243 \sqrt {3} b^2 x^2 \sqrt {27 (a-b x)^2+4} a-30 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+729 b^4 x^4+324 b^2 x^2+81 \sqrt {3} b^3 x^3 \sqrt {27 (a-b x)^2+4}+30 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+14\right )}{27 a^2-54 b x a-3 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+27 b^2 x^2+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+2}-\frac {(a-b x) \sqrt {27 (a-b x)^2+4} \left (\left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^4+288\right )}{(b x-a) \left (27 (a-b x)^2-3 \sqrt {3} a \sqrt {27 (a-b x)^2+4}+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+4\right )}}{576\ 2^{2/3} 3^{5/6} b \sqrt [3]{-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}}}+c_1\right \},\left \{y(x)\to \frac {24 \sqrt [3]{2} 3^{2/3} \left (1+i \sqrt {3}\right ) \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^{2/3} \left (9 \sqrt {3} a-9 \sqrt {3} b x+\sqrt {27 (a-b x)^2+4}\right )+\frac {12 \sqrt {3} \left (1-i \sqrt {3}\right ) \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right ) \left (729 a^4-2916 b x a^3-81 \sqrt {3} \sqrt {27 (a-b x)^2+4} a^3+4374 b^2 x^2 a^2+243 \sqrt {3} b x \sqrt {27 (a-b x)^2+4} a^2+324 a^2-2916 b^3 x^3 a-648 b x a-243 \sqrt {3} b^2 x^2 \sqrt {27 (a-b x)^2+4} a-30 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+729 b^4 x^4+324 b^2 x^2+81 \sqrt {3} b^3 x^3 \sqrt {27 (a-b x)^2+4}+30 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+14\right )}{27 a^2-54 b x a-3 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+27 b^2 x^2+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+2}+\frac {\left (1-i \sqrt {3}\right ) (a-b x) \sqrt {27 (a-b x)^2+4} \left (\left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^4+288\right )}{(b x-a) \left (27 (a-b x)^2-3 \sqrt {3} a \sqrt {27 (a-b x)^2+4}+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+4\right )}}{1152\ 2^{2/3} 3^{5/6} b \sqrt [3]{-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}}}+c_1\right \},\left \{y(x)\to \frac {24 \sqrt [3]{2} 3^{2/3} \left (1-i \sqrt {3}\right ) \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^{2/3} \left (9 \sqrt {3} a-9 \sqrt {3} b x+\sqrt {27 (a-b x)^2+4}\right )+\frac {12 \sqrt {3} \left (1+i \sqrt {3}\right ) \left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right ) \left (729 a^4-2916 b x a^3-81 \sqrt {3} \sqrt {27 (a-b x)^2+4} a^3+4374 b^2 x^2 a^2+243 \sqrt {3} b x \sqrt {27 (a-b x)^2+4} a^2+324 a^2-2916 b^3 x^3 a-648 b x a-243 \sqrt {3} b^2 x^2 \sqrt {27 (a-b x)^2+4} a-30 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+729 b^4 x^4+324 b^2 x^2+81 \sqrt {3} b^3 x^3 \sqrt {27 (a-b x)^2+4}+30 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+14\right )}{27 a^2-54 b x a-3 \sqrt {3} \sqrt {27 (a-b x)^2+4} a+27 b^2 x^2+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+2}+\frac {\left (1+i \sqrt {3}\right ) (a-b x) \sqrt {27 (a-b x)^2+4} \left (\left (-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}\right )^4+288\right )}{(b x-a) \left (27 (a-b x)^2-3 \sqrt {3} a \sqrt {27 (a-b x)^2+4}+3 \sqrt {3} b x \sqrt {27 (a-b x)^2+4}+4\right )}}{1152\ 2^{2/3} 3^{5/6} b \sqrt [3]{-9 a+9 b x+\sqrt {3} \sqrt {27 (a-b x)^2+4}}}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.065 (sec), leaf count = 268
\[ \left \{ y \left ( x \right ) =\int \!{{\frac {i}{12}} \left ( \left ( -\sqrt {3}+i \right ) \left ( 108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12} \right ) ^{{\frac {2}{3}}}-12\,i-12\,\sqrt {3} \right ) {\frac {1}{\sqrt [3]{108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{{\frac {i}{12}} \left ( \left ( \sqrt {3}+i \right ) \left ( 108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12} \right ) ^{{\frac {2}{3}}}-12\,i+12\,\sqrt {3} \right ) {\frac {1}{\sqrt [3]{108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{\frac {1}{6} \left ( \left ( 108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12} \right ) ^{{\frac {2}{3}}}-12 \right ) {\frac {1}{\sqrt [3]{108\,bx-108\,a+12\,\sqrt {81\,{b}^{2}{x}^{2}-162\,abx+81\,{a}^{2}+12}}}}}\,{\rm d}x+{\it \_C1} \right \} \] Mathematica raw input
DSolve[a - b*x + y'[x] + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-24*2^(1/3)*3^(2/3)*(9*Sqrt[3]*a - 9*Sqrt[3]*b*x + Sqrt[4 + 27*(a - b*x)^2])*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(2/3) - (12*Sqrt[3]*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])*(14 + 324*a^2 + 729*a^4 - 648*a*b*x - 2916*a^3*b*x + 324*b^2*x^2 + 4374*a^2*b^2*x^2 - 2916*a*b^3*x^3 + 729*b^4*x^4 - 30*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] - 81*Sqrt[3]*a^3*Sqrt[4 + 27*(a - b*x)^2] + 30*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2] + 243*Sqrt[3]*a^2*b*x*Sqrt[4 + 27*(a - b*x)^2] - 243*Sqrt[3]*a*b^2*x^2*Sqrt[4 + 27*(a - b*x)^2] + 81*Sqrt[3]*b^3*x^3*Sqrt[4 + 27*(a - b*x)^2]))/(2 + 27*a^2 - 54*a*b*x + 27*b^2*x^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2]) - ((a - b*x)*Sqrt[4 + 27*(a - b*x)^2]*(288 + (-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^4))/((-a + b*x)*(4 + 27*(a - b*x)^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2])))/(576*2^(2/3)*3^(5/6)*b*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(1/3)) + C[1]}, {y[x] -> (24*2^(1/3)*3^(2/3)*(1 + I*Sqrt[3])*(9*Sqrt[3]*a - 9*Sqrt[3]*b*x + Sqrt[4 + 27*(a - b*x)^2])*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(2/3) + (12*Sqrt[3]*(1 - I*Sqrt[3])*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])*(14 + 324*a^2 + 729*a^4 - 648*a*b*x - 2916*a^3*b*x + 324*b^2*x^2 + 4374*a^2*b^2*x^2 - 2916*a*b^3*x^3 + 729*b^4*x^4 - 30*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] - 81*Sqrt[3]*a^3*Sqrt[4 + 27*(a - b*x)^2] + 30*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2] + 243*Sqrt[3]*a^2*b*x*Sqrt[4 + 27*(a - b*x)^2] - 243*Sqrt[3]*a*b^2*x^2*Sqrt[4 + 27*(a - b*x)^2] + 81*Sqrt[3]*b^3*x^3*Sqrt[4 + 27*(a - b*x)^2]))/(2 + 27*a^2 - 54*a*b*x + 27*b^2*x^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2]) + ((1 - I*Sqrt[3])*(a - b*x)*Sqrt[4 + 27*(a - b*x)^2]*(288 + (-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^4))/((-a + b*x)*(4 + 27*(a - b*x)^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2])))/(1152*2^(2/3)*3^(5/6)*b*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(1/3)) + C[1]}, {y[x] -> (24*2^(1/3)*3^(2/3)*(1 - I*Sqrt[3])*(9*Sqrt[3]*a - 9*Sqrt[3]*b*x + Sqrt[4 + 27*(a - b*x)^2])*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(2/3) + (12*Sqrt[3]*(1 + I*Sqrt[3])*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])*(14 + 324*a^2 + 729*a^4 - 648*a*b*x - 2916*a^3*b*x + 324*b^2*x^2 + 4374*a^2*b^2*x^2 - 2916*a*b^3*x^3 + 729*b^4*x^4 - 30*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] - 81*Sqrt[3]*a^3*Sqrt[4 + 27*(a - b*x)^2] + 30*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2] + 243*Sqrt[3]*a^2*b*x*Sqrt[4 + 27*(a - b*x)^2] - 243*Sqrt[3]*a*b^2*x^2*Sqrt[4 + 27*(a - b*x)^2] + 81*Sqrt[3]*b^3*x^3*Sqrt[4 + 27*(a - b*x)^2]))/(2 + 27*a^2 - 54*a*b*x + 27*b^2*x^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2]) + ((1 + I*Sqrt[3])*(a - b*x)*Sqrt[4 + 27*(a - b*x)^2]*(288 + (-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^4))/((-a + b*x)*(4 + 27*(a - b*x)^2 - 3*Sqrt[3]*a*Sqrt[4 + 27*(a - b*x)^2] + 3*Sqrt[3]*b*x*Sqrt[4 + 27*(a - b*x)^2])))/(1152*2^(2/3)*3^(5/6)*b*(-9*a + 9*b*x + Sqrt[3]*Sqrt[4 + 27*(a - b*x)^2])^(1/3)) + C[1]}}
Maple raw input
dsolve(diff(y(x),x)^3+diff(y(x),x)+a-b*x = 0, y(x),'implicit')
Maple raw output
y(x) = Int(1/12*I/(108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(1/3)*((-3^(1/2)+I)*(108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(2/3)-12*I-12*3^(1/2)),x)+_C1, y(x) = Int(1/12*I/(108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(1/3)*((3^(1/2)+I)*(108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(2/3)-12*I+12*3^(1/2)),x)+_C1, y(x) = Int(1/6*((108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(2/3)-12)/(108*b*x-108*a+12*(81*b^2*x^2-162*a*b*x+81*a^2+12)^(1/2))^(1/3),x)+_C1