Internal problem ID [4254]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1029.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
\[ \boxed {{y^{\prime }}^{3}-a x y^{\prime }=-x^{3}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 252
dsolve(diff(y(x),x)^3-a*x*diff(y(x),x)+x^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (\int \left (\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )-\frac {12 a \left (1+i \sqrt {3}\right ) x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \left (\left (1+i \sqrt {3}\right ) \left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}-\frac {12 a \left (-1+i \sqrt {3}\right ) x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {2}{3}}+12 a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 166.72 (sec). Leaf size: 349
DSolve[(y'[x])^3 -a*x*y'[x]+x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \int _1^x\frac {2 \sqrt [3]{3} a K[1]+\sqrt [3]{2} \left (\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3\right )^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3}}dK[1]+c_1 \\ y(x)\to \int _1^x\frac {i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (2 \sqrt {81 K[2]^6-12 a^3 K[2]^3}-18 K[2]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) a K[2]}{12 \sqrt [3]{\sqrt {81 K[2]^6-12 a^3 K[2]^3}-9 K[2]^3}}dK[2]+c_1 \\ y(x)\to \int _1^x\frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {81 K[3]^6-12 a^3 K[3]^3}-18 K[3]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) a K[3]}{12 \sqrt [3]{\sqrt {81 K[3]^6-12 a^3 K[3]^3}-9 K[3]^3}}dK[3]+c_1 \\ \end{align*}