ODE
\[ x^3 y'(x)^2+x y'(x)-y(x)=0 \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 2.04036 (sec), leaf count = 2799
\[\left \{\left \{y(x)\to -\frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}} x+\sqrt {3} \sqrt {\frac {96 \sqrt {3} \left (x^2+4 c_1{}^2\right )}{x^3 c_1{}^2 \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}}}-\frac {\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}-\frac {x \left (x^2+864 c_1{}^2\right )}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1{}^2}} x-36}{24 x}\right \},\left \{y(x)\to \frac {-\sqrt {3} \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}} x+\sqrt {3} \sqrt {\frac {96 \sqrt {3} \left (x^2+4 c_1{}^2\right )}{x^3 c_1{}^2 \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}}}-\frac {\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}-\frac {x \left (x^2+864 c_1{}^2\right )}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1{}^2}} x+36}{24 x}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}} x-\sqrt {3} \sqrt {-\frac {96 \sqrt {3} \left (x^2+4 c_1{}^2\right )}{x^3 c_1{}^2 \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}}}-\frac {\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}-\frac {x \left (x^2+864 c_1{}^2\right )}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1{}^2}} x+36}{24 x}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}} x+\sqrt {3} \sqrt {-\frac {96 \sqrt {3} \left (x^2+4 c_1{}^2\right )}{x^3 c_1{}^2 \sqrt {-\frac {-\frac {c_1{}^4 x^6}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}-\frac {864 c_1{}^6 x^4}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+c_1{}^2 x^3-48 c_1{}^4 x-\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}}}-\frac {\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}{x^3 c_1{}^4}-\frac {x \left (x^2+864 c_1{}^2\right )}{\sqrt [3]{93312 x^5 c_1{}^{10}+2160 x^7 c_1{}^8-x^9 c_1{}^6+48 \sqrt {3} \sqrt {-x^{10} c_1{}^{14} \left (x^2-108 c_1{}^2\right ){}^3}}}+\frac {96}{x^2}-\frac {2}{c_1{}^2}} x+36}{24 x}\right \}\right \}\]
Maple ✗
cpu = 0. (sec), leaf count = 0 , exception
time expired
Mathematica raw input
DSolve[-y[x] + x*y'[x] + x^3*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -1/24*(-36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) + (96*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/x}, {y[x] -> (36 - Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) + (96*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*x)}, {y[x] -> (36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] - Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) - (96*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*x)}, {y[x] -> (36 + Sqrt[3]*x*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))] + Sqrt[3]*x*Sqrt[96/x^2 - 2/C[1]^2 - (x*(x^2 + 864*C[1]^2))/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3)/(x^3*C[1]^4) - (96*Sqrt[3]*(x^2 + 4*C[1]^2))/(x^3*C[1]^2*Sqrt[-((x^3*C[1]^2 - 48*x*C[1]^4 - (x^6*C[1]^4)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (864*x^4*C[1]^6)/(-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3) - (-(x^9*C[1]^6) + 2160*x^7*C[1]^8 + 93312*x^5*C[1]^10 + 48*Sqrt[3]*Sqrt[-(x^10*C[1]^14*(x^2 - 108*C[1]^2)^3)])^(1/3))/(x^3*C[1]^4))])])/(24*x)}}
Maple raw input
dsolve(x^3*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0, y(x))
Maple raw output
\verbtime expired||