Internal problem ID [3735]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1018.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}-\left (a +b y+c y^{2}\right ) f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 191
dsolve(diff(y(x),x)^3 = (a+b*y(x)+c*y(x)^2)*f(x),y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (\left (a +b y \relax (x )+c y \relax (x )^{2}\right ) f \left (\textit {\_a} \right )\right )^{\frac {1}{3}}}{\left (a +b y \relax (x )+c y \relax (x )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +c_{1} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} +\int _{}^{x}\frac {\left (\left (a +b y \relax (x )+c y \relax (x )^{2}\right ) f \left (\textit {\_a} \right )\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 \left (a +b y \relax (x )+c y \relax (x )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +c_{1} = 0 \\ \int _{}^{y \relax (x )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (\left (a +b y \relax (x )+c y \relax (x )^{2}\right ) f \left (\textit {\_a} \right )\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{2 \left (a +b y \relax (x )+c y \relax (x )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 62.605 (sec). Leaf size: 405
DSolve[(y'[x])^3 ==(a+b y[x]+c y[x]^2) f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {(2 \text {$\#$1} c+b) \sqrt [3]{\frac {c (\text {$\#$1} (\text {$\#$1} c+b)+a)}{4 a c-b^2}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(2 \text {$\#$1} c+b)^2}{b^2-4 a c}\right )}{\sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\&\right ]\left [\int _1^x\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {(2 \text {$\#$1} c+b) \sqrt [3]{\frac {c (\text {$\#$1} (\text {$\#$1} c+b)+a)}{4 a c-b^2}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(2 \text {$\#$1} c+b)^2}{b^2-4 a c}\right )}{\sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\&\right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {(2 \text {$\#$1} c+b) \sqrt [3]{\frac {c (\text {$\#$1} (\text {$\#$1} c+b)+a)}{4 a c-b^2}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(2 \text {$\#$1} c+b)^2}{b^2-4 a c}\right )}{\sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\&\right ]\left [\int _1^x(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ] \\ y(x)\to \frac {\sqrt {b^2-4 a c}-b}{2 c} \\ y(x)\to -\frac {\sqrt {b^2-4 a c}+b}{2 c} \\ \end{align*}