Internal problem ID [14396]
Internal file name [OUTPUT/14077_Monday_March_11_2024_01_33_31_AM_79828555/index.tex]
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.
The type(s) of ODE detected by this program : "dAlembert"
Maple gives the following as the ode type
[_dAlembert]
\[ \boxed {y-t {y^{\prime }}^{2}-3 {y^{\prime }}^{2}+2 {y^{\prime }}^{3}=0} \]
Let \(p=y^{\prime }\) the ode becomes \begin {align*} 2 p^{3}-t \,p^{2}-3 p^{2}+y = 0 \end {align*}
Solving for \(y\) from the above results in \begin {align*} y &= -2 p^{3}+t \,p^{2}+3 p^{2}\tag {1A} \end {align*}
This has the form \begin {align*} y=tf(p)+g(p)\tag {*} \end {align*}
Where \(f,g\) are functions of \(p=y'(t)\). The above ode is dAlembert ode which is now solved. Taking derivative of (*) w.r.t. \(t\) gives \begin {align*} p &= f+(t f'+g') \frac {dp}{dt}\\ p-f &= (t f'+g') \frac {dp}{dt}\tag {2} \end {align*}
Comparing the form \(y=t f + g\) to (1A) shows that \begin {align*} f &= p^{2}\\ g &= -2 p^{3}+3 p^{2} \end {align*}
Hence (2) becomes \begin {align*} -p^{2}+p = \left (-6 p^{2}+2 t p +6 p \right ) p^{\prime }\left (t \right )\tag {2A} \end {align*}
The singular solution is found by setting \(\frac {dp}{dt}=0\) in the above which gives \begin {align*} -p^{2}+p = 0 \end {align*}
Solving for \(p\) from the above gives \begin {align*} p&=0\\ p&=1 \end {align*}
Substituting these in (1A) gives \begin {align*} y&=0\\ y&=t +1 \end {align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}t}}\neq 0\). From eq. (2A). This results in \begin {align*} p^{\prime }\left (t \right ) = \frac {-p \left (t \right )^{2}+p \left (t \right )}{-6 p \left (t \right )^{2}+2 p \left (t \right ) t +6 p \left (t \right )}\tag {3} \end {align*}
This ODE is now solved for \(p \left (t \right )\).
Inverting the above ode gives \begin {align*} \frac {d}{d p}t \left (p \right ) = \frac {-6 p^{2}+2 t \left (p \right ) p +6 p}{-p^{2}+p}\tag {4} \end {align*}
This ODE is now solved for \(t \left (p \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d p}t \left (p \right ) + p(p)t \left (p \right ) &= q(p) \end {align*}
Where here \begin {align*} p(p) &=\frac {2}{p -1}\\ q(p) &=-\frac {2 \left (-3 p +3\right )}{p -1} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d p}t \left (p \right )+\frac {2 t \left (p \right )}{p -1} = -\frac {2 \left (-3 p +3\right )}{p -1} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p -1}d p} \\ &= \left (p -1\right )^{2} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}}\left ( \mu t\right ) &= \left (\mu \right ) \left (-\frac {2 \left (-3 p +3\right )}{p -1}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}p}} \left (\left (p -1\right )^{2} t\right ) &= \left (\left (p -1\right )^{2}\right ) \left (-\frac {2 \left (-3 p +3\right )}{p -1}\right )\\ \mathrm {d} \left (\left (p -1\right )^{2} t\right ) &= \left (6 \left (p -1\right )^{2}\right )\, \mathrm {d} p \end {align*}
Integrating gives \begin {align*} \left (p -1\right )^{2} t &= \int {6 \left (p -1\right )^{2}\,\mathrm {d} p}\\ \left (p -1\right )^{2} t &= 2 \left (p -1\right )^{3} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\left (p -1\right )^{2}\) results in \begin {align*} t \left (p \right ) &= 2 p -2+\frac {c_{1}}{\left (p -1\right )^{2}} \end {align*}
Now we need to eliminate \(p\) between the above and (1A). One way to do this is by solving (1) for \(p\). This results in \begin {align*} p&=\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}+\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}\\ p&=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2}\\ p&=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2} \end {align*}
Substituting the above in the solution for \(t\) found above gives \begin{align*} t&=\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{3}+\frac {\left (t +3\right )^{2}}{3 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {36 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t -3\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}+\left (t +3\right )^{2}\right )}^{2}} \\ t&=-1-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (i \sqrt {3}-1\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (2 t -6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (1+i \sqrt {3}\right )\right )}^{2}} \\ t&=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (1+i \sqrt {3}\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (-2 t +6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (i \sqrt {3}-1\right )\right )}^{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \tag{2} y &= t +1 \\ \tag{3} t &= \frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{3}+\frac {\left (t +3\right )^{2}}{3 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {36 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t -3\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}+\left (t +3\right )^{2}\right )}^{2}} \\ \tag{4} t &= -1-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (i \sqrt {3}-1\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (2 t -6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (1+i \sqrt {3}\right )\right )}^{2}} \\ \tag{5} t &= -\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (1+i \sqrt {3}\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (-2 t +6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (i \sqrt {3}-1\right )\right )}^{2}} \\ \end{align*}
Verification of solutions
\[ y = 0 \] Verified OK.
\[ y = t +1 \] Verified OK.
\[ t = \frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{3}+\frac {\left (t +3\right )^{2}}{3 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {36 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t -3\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}+\left (t +3\right )^{2}\right )}^{2}} \] Verified OK.
\[ t = -1-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (i \sqrt {3}-1\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (2 t -6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (1+i \sqrt {3}\right )\right )}^{2}} \] Verified OK.
\[ t = -\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {t}{3}-1+\frac {i \sqrt {3}\, \left (-\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (t +3\right )^{2}\right )}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}}+\frac {144 c_{1} \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}}{{\left (\left (1+i \sqrt {3}\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{2}/{3}}+\left (-2 t +6\right ) \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +t^{3}+9 t^{2}\right )}\right )^{{1}/{3}}-\left (t +3\right )^{2} \left (i \sqrt {3}-1\right )\right )}^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y-t {y^{\prime }}^{2}-3 {y^{\prime }}^{2}+2 {y^{\prime }}^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}+\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}, y^{\prime }=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}+\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{12}-\frac {\left (t +3\right )^{2}}{12 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}+\frac {t}{6}+\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}{6}-\frac {\left (t +3\right )^{2}}{6 \left (-54 y+t^{3}+9 t^{2}+27 t +27+6 \sqrt {81 y^{2}-81 y-81 y t -3 y t^{3}-27 y t^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: *** Sublevel 2 *** Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying dAlembert <- dAlembert successful`
✓ 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*}