74.7.54 problem 59
Internal
problem
ID
[16131]
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
Date
solved
:
Tuesday, January 28, 2025 at 08:50:24 AM
CAS
classification
:
[_dAlembert]
\begin{align*} y&=t {y^{\prime }}^{2}+3 {y^{\prime }}^{2}-2 {y^{\prime }}^{3} \end{align*}
✓ Solution by Maple
Time used: 0.097 (sec). Leaf size: 734
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 &= 0 \\
y &= -\frac {\left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{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 )^{{1}/{3}}+t^{2}\right ) {\left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{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 )^{{1}/{3}}+t^{2}\right )}^{2}}{108 t^{3}+648 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-5832 c_{1}} \\
y &= \frac {{\left (i \sqrt {3}\, \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{2}/{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 )^{{2}/{3}}-2 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{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 )^{{1}/{3}}\right )}^{2} \left (i \sqrt {3}\, \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{2}/{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 )^{{2}/{3}}+4 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{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 )^{{1}/{3}}\right )}{864 t^{3}+5184 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-46656 c_{1}} \\
y &= \frac {\left (i \sqrt {3}\, t^{2}-i \sqrt {3}\, \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{2}/{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 )^{{1}/{3}}-12 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{1}/{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{2}/{3}}\right )^{2} \left (i \sqrt {3}\, t^{2}-i \sqrt {3}\, \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{2}/{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 )^{{1}/{3}}+6 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{1}/{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{{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: 126.170 (sec). Leaf size: 875
DSolve[y[t]==t*D[y[t],t]^2+3*D[y[t],t]^2-2*D[y[t],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*}