Internal problem ID [3759]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1046.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}+\left (-3 x +1\right ) \left (y^{\prime }\right )^{2}-x \left (-3 x +1\right ) y^{\prime }-1-x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 496
dsolve(diff(y(x),x)^3+(1-3*x)*diff(y(x),x)^2-x*(1-3*x)*diff(y(x),x)-1-x^3 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \int -\frac {i \left (\sqrt {3}\, \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}-i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+12 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}} x +12 \sqrt {3}\, x -4 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}+12 i x -4 \sqrt {3}-4 i\right )}{12 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ y \relax (x ) = \int \frac {i \left (\sqrt {3}\, \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+12 \sqrt {3}\, x -4 \sqrt {3}+i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}-12 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}} x +4 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-12 i x +4 i\right )}{12 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ y \relax (x ) = \int \frac {\left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 178.017 (sec). Leaf size: 344
DSolve[(y'[x])^3+(1-3 x)(y'[x])^2-x(1-3 x)y'[x]-1 -x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \int _1^x\frac {1}{6} \left (6 K[1]-2^{2/3} \sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}+\frac {2 \sqrt [3]{2} (3 K[1]-1)}{\sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}}-2\right )dK[1]+c_1 \\ y(x)\to \int _1^x\frac {1}{12} \left (\frac {4 \sqrt [3]{-2} (1-3 K[2])}{\sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}}+12 K[2]-2 (-2)^{2/3} \sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}-4\right )dK[2]+c_1 \\ y(x)\to \int _1^x\left (K[3]+\frac {1}{3} \sqrt [3]{-\frac {1}{2}} \sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}+\frac {(-1)^{2/3} \sqrt [3]{2} (3 K[3]-1)}{3 \sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}}-\frac {1}{3}\right )dK[3]+c_1 \\ \end{align*}