Internal problem ID [14397]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number: 59.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_dAlembert]
\[ \boxed {y-t {y^{\prime }}^{2}-3 {y^{\prime }}^{2}+2 {y^{\prime }}^{3}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 742
dsolve(y(t)=t*diff(y(t),t)^2+3*diff(y(t),t)^2-2*diff(y(t),t)^3,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= -\frac {{\left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+\left (t +6\right ) \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}\right )}^{2} \left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+\left (-2 t -3\right ) \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}\right )}{108 t^{3}+648 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-5832 c_{1}} \\ y \left (t \right ) &= \frac {{\left (i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, t^{2}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}-2 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}-12 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}\right )}^{2} \left (i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, t^{2}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+4 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}+6 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}\right )}{864 t^{3}+5184 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-46656 c_{1}} \\ y \left (t \right ) &= \frac {\left (i \sqrt {3}\, t^{2}-i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+t^{2}-2 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}-12 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )^{2} \left (i \sqrt {3}\, t^{2}-i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+t^{2}+4 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+6 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{864 t^{3}+5184 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-46656 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 135.31 (sec). Leaf size: 875
DSolve[y[t]==t*y'[t]^2+3*y'[t]^2-2*y'[t]^3,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{12} \left (-\frac {2 t^3}{\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}}-2 t \left (-6+\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}\right )-\left (-t^3+6 \sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+108+54 c_1\right ){}^{2/3}+t^2-\frac {t^4}{\left (-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )\right ){}^{2/3}}+36+12 c_1\right ) \\ y(t)\to \frac {1}{24} \left (\frac {2 \left (1-i \sqrt {3}\right ) t^3}{\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}}+2 t \left (i \sqrt {3} \sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}+\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}+12\right )-i \sqrt {3} \left (-t^3+6 \sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+108+54 c_1\right ){}^{2/3}+\left (-t^3+6 \sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+108+54 c_1\right ){}^{2/3}+2 t^2+\frac {\left (1+i \sqrt {3}\right ) t^4}{\left (-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )\right ){}^{2/3}}+72+24 c_1\right ) \\ y(t)\to \frac {1}{24} \left (\frac {2 \left (1+i \sqrt {3}\right ) t^3}{\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}}+2 t \left (-i \sqrt {3} \sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}+\sqrt [3]{-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )}+12\right )+i \sqrt {3} \left (-t^3+6 \sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+108+54 c_1\right ){}^{2/3}+\left (-t^3+6 \sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+108+54 c_1\right ){}^{2/3}+2 t^2+\frac {\left (1-i \sqrt {3}\right ) t^4}{\left (-t^3+6 \left (\sqrt {3} \sqrt {(2+c_1) \left (-t^3+54+27 c_1\right )}+18+9 c_1\right )\right ){}^{2/3}}+72+24 c_1\right ) \\ \end{align*}