Internal problem ID [13022]
Internal file name [OUTPUT/11675_Wednesday_November_08_2023_03_28_27_AM_88496519/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.9 page 133
Problem number: 18.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "linear", "exactWithIntegrationFactor", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }-\frac {y}{\sqrt {t^{3}-3}}=t} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(t)y &= q(t) \end {align*}
Where here \begin {align*} p(t) &=-\frac {1}{\sqrt {t^{3}-3}}\\ q(t) &=t \end {align*}
Hence the ode is \begin {align*} y^{\prime }-\frac {y}{\sqrt {t^{3}-3}} = t \end {align*}
The integrating factor \(\mu \) is \[ \mu = {\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}}\left ( \mu y\right ) &= \left (\mu \right ) \left (t\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left ({\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} y\right ) &= \left ({\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t}\right ) \left (t\right )\\ \mathrm {d} \left ({\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} y\right ) &= \left (t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}\right )\, \mathrm {d} t \end {align*}
Integrating gives \begin {align*} {\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} y &= \int {t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}\,\mathrm {d} t}\\ {\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} y &= \int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t}\) results in \begin {align*} y &= {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \left (\int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t \right )+c_{1} {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \end {align*}
which simplifies to \begin {align*} y &= {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \left (\int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \left (\int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \left (\int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t +c_{1} \right ) \] Verified OK.
Writing the ode as \begin {align*} y^{\prime }&=\frac {t \sqrt {t^{3}-3}+y}{\sqrt {t^{3}-3}}\\ y^{\prime }&= \omega \left ( t,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{t}+\omega \left ( \eta _{y}-\xi _{t}\right ) -\omega ^{2}\xi _{y}-\omega _{t}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is known. It is of type linear. Therefore we do not need
to solve the PDE (A), and can just use the lookup table shown below to find \(\xi ,\eta \)
|
ODE class |
Form |
\(\xi \) |
\(\eta \) |
|
linear ode |
\(y'=f(x) y(x) +g(x)\) |
\(0\) |
\(e^{\int fdx}\) |
|
separable ode |
\(y^{\prime }=f\left ( x\right ) g\left ( y\right ) \) |
\(\frac {1}{f}\) |
\(0\) |
|
quadrature ode |
\(y^{\prime }=f\left ( x\right ) \) |
\(0\) |
\(1\) |
|
quadrature ode |
\(y^{\prime }=g\left ( y\right ) \) |
\(1\) |
\(0\) |
|
homogeneous ODEs of Class A |
\(y^{\prime }=f\left ( \frac {y}{x}\right ) \) |
\(x\) |
\(y\) |
|
homogeneous ODEs of Class C |
\(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\) |
\(1\) |
\(-\frac {b}{c}\) |
|
homogeneous class D |
\(y^{\prime }=\frac {y}{x}+g\left ( x\right ) F\left (\frac {y}{x}\right ) \) |
\(x^{2}\) |
\(xy\) |
|
First order special form ID 1 |
\(y^{\prime }=g\left ( x\right ) e^{h\left (x\right ) +by}+f\left ( x\right ) \) |
\(\frac {e^{-\int bf\left ( x\right )dx-h\left ( x\right ) }}{g\left ( x\right ) }\) |
\(\frac {f\left ( x\right )e^{-\int bf\left ( x\right ) dx-h\left ( x\right ) }}{g\left ( x\right ) }\) |
|
polynomial type ode |
\(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\) |
\(\frac {a_{1}b_{2}x-a_{2}b_{1}x-b_{1}c_{2}+b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) |
\(\frac {a_{1}b_{2}y-a_{2}b_{1}y-a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}\) |
|
Bernoulli ode |
\(y^{\prime }=f\left ( x\right ) y+g\left ( x\right ) y^{n}\) |
\(0\) |
\(e^{-\int \left ( n-1\right ) f\left ( x\right ) dx}y^{n}\) |
|
Reduced Riccati |
\(y^{\prime }=f_{1}\left ( x\right ) y+f_{2}\left ( x\right )y^{2}\) |
\(0\) |
\(e^{-\int f_{1}dx}\) |
|
|
|||
|
|
|||
The above table shows that \begin {align*} \xi \left (t,y\right ) &=0\\ \tag {A1} \eta \left (t,y\right ) &={\mathrm e}^{\frac {2 i 3^{\frac {5}{6}} \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\, \sqrt {i \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}}{3}, \sqrt {-\frac {i 3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\right )}{3 \sqrt {t^{3}-3}}} \end {align*}
The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( t,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d t}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial t} + \eta \frac {\partial }{\partial y}\right ) S(t,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Since \(\xi =0\) then in this special case \begin {align*} R = t \end {align*}
\(S\) is found from \begin {align*} S &= \int { \frac {1}{\eta }} dy\\ &= \int { \frac {1}{{\mathrm e}^{\frac {2 i 3^{\frac {5}{6}} \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\, \sqrt {i \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}}{3}, \sqrt {-\frac {i 3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\right )}{3 \sqrt {t^{3}-3}}}}} dy \end {align*}
Which results in \begin {align*} S&= {\mathrm e}^{-\frac {2 i \operatorname {EllipticF}\left (\frac {\sqrt {6}\, \sqrt {-i \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}+2 t \right ) 3^{\frac {1}{6}}}}{6}, \sqrt {-\frac {i 3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\right ) \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\, \sqrt {i \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, 3^{\frac {5}{6}}}{3 \sqrt {t^{3}-3}}} y \end {align*}
Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{t} + \omega (t,y) S_{y} }{ R_{t} + \omega (t,y) R_{y} }\tag {2} \end {align*}
Where in the above \(R_{t},R_{y},S_{t},S_{y}\) are all partial derivatives and \(\omega (t,y)\) is the right hand side of the original ode given by \begin {align*} \omega (t,y) &= \frac {t \sqrt {t^{3}-3}+y}{\sqrt {t^{3}-3}} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{t} &= 1\\ R_{y} &= 0\\ S_{t} &= -\frac {12 \,{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} \left (i 3^{\frac {11}{12}} t -2 i 3^{\frac {7}{12}} t^{2}-3 \,3^{\frac {5}{12}} t +3 \,3^{\frac {3}{4}}+3 i 3^{\frac {1}{4}}\right ) y}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}\\ S_{y} &= {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= \frac {24 \left (\frac {\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \left (t \sqrt {t^{3}-3}+y \right ) \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}}{24}+\left (i 3^{\frac {7}{12}} t^{2}-\frac {i 3^{\frac {11}{12}} t}{2}-\frac {3 i 3^{\frac {1}{4}}}{2}+\frac {3 \,3^{\frac {5}{12}} t}{2}-\frac {3 \,3^{\frac {3}{4}}}{2}\right ) y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\tag {2A} \end {align*}
We now need to express the RHS as function of \(R\) only. This is done by solving for \(t,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= \frac {\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \left (\sqrt {R^{3}-3}\, {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i R}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {R^{3}-3}}} R +S \left (R \right )\right ) \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}+24 \left (i 3^{\frac {7}{12}} R^{2}-\frac {i 3^{\frac {11}{12}} R}{2}-\frac {3 i 3^{\frac {1}{4}}}{2}+\frac {3 \,3^{\frac {5}{12}} R}{2}-\frac {3 \,3^{\frac {3}{4}}}{2}\right ) S \left (R \right )}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {R^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}} \end {align*}
The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives \begin {align*} S \left (R \right ) = \left (\int R \,{\mathrm e}^{-\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i R}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i R}-\left (\int \frac {12 i 3^{\frac {11}{12}} R -24 i 3^{\frac {7}{12}} R^{2}-36 \,3^{\frac {5}{12}} R -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {R^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}}d R \right ) \sqrt {R^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {R^{3}-3}}}d R +c_{1} \right ) {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} R -24 i 3^{\frac {7}{12}} R^{2}-36 \,3^{\frac {5}{12}} R -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} R -6 \,3^{\frac {2}{3}} R}\, \sqrt {3^{\frac {1}{3}}-R}\, \sqrt {R^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} R}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i R}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d R}\tag {4} \end {align*}
To complete the solution, we just need to transform (4) back to \(t,y\) coordinates. This results in \begin {align*} y \,{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} = \left (\int t \,{\mathrm e}^{-\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}-\left (\int \frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}}d t \right ) \sqrt {t^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d t} \end {align*}
Which simplifies to \begin {align*} y \,{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} = \left (\int t \,{\mathrm e}^{-\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}-\left (\int \frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}}d t \right ) \sqrt {t^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d t} \end {align*}
Which gives \begin {align*} y = {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d t} \left (\int t \,{\mathrm e}^{\frac {-i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}+\left (\int \frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}}d t \right ) \sqrt {t^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d t} \left (\int t \,{\mathrm e}^{\frac {-i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}+\left (\int \frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}}d t \right ) \sqrt {t^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\int -\frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}d t} \left (\int t \,{\mathrm e}^{\frac {-i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}+\left (\int \frac {12 i 3^{\frac {11}{12}} t -24 i 3^{\frac {7}{12}} t^{2}-36 \,3^{\frac {5}{12}} t -\sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}+36 \,3^{\frac {3}{4}}+36 i 3^{\frac {1}{4}}}{\sqrt {18-6 i \sqrt {3}+6 i 3^{\frac {1}{6}} t -6 \,3^{\frac {2}{3}} t}\, \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {t^{3}-3}\, \sqrt {18+6 i \sqrt {3}+12 i 3^{\frac {1}{6}} t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}}d t \right ) \sqrt {t^{3}-3}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\frac {i \operatorname {EllipticF}\left (\frac {\sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}\, 3^{\frac {7}{12}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right ) \sqrt {3^{\frac {1}{3}}-t}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \sqrt {2 \,3^{\frac {5}{6}}-2 i 3^{\frac {1}{3}}-4 i t}}{\sqrt {i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}\, \sqrt {t^{3}-3}}} \] Verified OK.
Entering Exact first order ODE solver. (Form one type)
To solve an ode of the form\begin {equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A} \end {equation} We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \] Hence\begin {equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B} \end {equation} Comparing (A,B) shows that\begin {align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end {align*}
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\] If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is \[ M(t,y) \mathop {\mathrm {d}t}+ N(t,y) \mathop {\mathrm {d}y}=0 \tag {1A} \] Therefore \begin {align*} \mathop {\mathrm {d}y} &= \left (\frac {y}{\sqrt {t^{3}-3}}+t\right )\mathop {\mathrm {d}t}\\ \left (-\frac {y}{\sqrt {t^{3}-3}}-t\right ) \mathop {\mathrm {d}t} + \mathop {\mathrm {d}y} &= 0 \tag {2A} \end {align*}
Comparing (1A) and (2A) shows that \begin {align*} M(t,y) &= -\frac {y}{\sqrt {t^{3}-3}}-t\\ N(t,y) &= 1 \end {align*}
The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied \[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial t} \] Using result found above gives \begin {align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (-\frac {y}{\sqrt {t^{3}-3}}-t\right )\\ &= -\frac {1}{\sqrt {t^{3}-3}} \end {align*}
And \begin {align*} \frac {\partial N}{\partial t} &= \frac {\partial }{\partial t} \left (1\right )\\ &= 0 \end {align*}
Since \(\frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial t}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let \begin {align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial y} - \frac {\partial N}{\partial t} \right ) \\ &=1\left ( \left ( -\frac {1}{\sqrt {t^{3}-3}}\right ) - \left (0 \right ) \right ) \\ &=-\frac {1}{\sqrt {t^{3}-3}} \end {align*}
Since \(A\) does not depend on \(y\), then it can be used to find an integrating factor. The integrating factor \(\mu \) is \begin {align*} \mu &= e^{ \int A \mathop {\mathrm {d}t} } \\ &= e^{\int -\frac {1}{\sqrt {t^{3}-3}}\mathop {\mathrm {d}t} } \end {align*}
The result of integrating gives \begin {align*} \mu &= e^{-\frac {2 i 3^{\frac {5}{6}} \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\, \sqrt {i \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-i \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}}{3}, \sqrt {-\frac {i 3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}}}\right )}{3 \sqrt {t^{3}-3}} } \\ &= {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}} \end {align*}
\(M\) and \(N\) are multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) for now so not to confuse them with the original \(M\) and \(N\). \begin {align*} \overline {M} &=\mu M \\ &= {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\left (-\frac {y}{\sqrt {t^{3}-3}}-t\right ) \\ &= \frac {\left (-t \sqrt {t^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}}{\sqrt {t^{3}-3}} \end {align*}
And \begin {align*} \overline {N} &=\mu N \\ &= {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\left (1\right ) \\ &= {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}} \end {align*}
Now a modified ODE is ontained from the original ODE, which is exact and can be solved. The modified ODE is \begin {align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}t}} &= 0 \\ \left (\frac {\left (-t \sqrt {t^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}}{\sqrt {t^{3}-3}}\right ) + \left ({\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}t}} &= 0 \end {align*}
The following equations are now set up to solve for the function \(\phi \left (t,y\right )\) \begin {align*} \frac {\partial \phi }{\partial t } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial y } &= \overline {N}\tag {2} \end {align*}
Integrating (1) w.r.t. \(t\) gives \begin{align*} \int \frac {\partial \phi }{\partial t} \mathop {\mathrm {d}t} &= \int \overline {M}\mathop {\mathrm {d}t} \\ \int \frac {\partial \phi }{\partial t} \mathop {\mathrm {d}t} &= \int \frac {\left (-t \sqrt {t^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}}{\sqrt {t^{3}-3}}\mathop {\mathrm {d}t} \\ \tag{3} \phi &= \int _{}^{t}\frac {\left (-\textit {\_a} \sqrt {\textit {\_a}^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a}+ f(y) \\ \end{align*} Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function of both \(t\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives \begin{equation} \tag{4} \frac {\partial \phi }{\partial y} = -\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} \right )+f'(y) \end{equation} But equation (2) says that \(\frac {\partial \phi }{\partial y} = {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\). Therefore equation (4) becomes \begin{equation} \tag{5} {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}} = -\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} \right )+f'(y) \end{equation} Solving equation (5) for \( f'(y)\) gives \[ f'(y) = \int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}} \] Integrating the above w.r.t \(y\) gives \begin{align*} \int f'(y) \mathop {\mathrm {d}y} &= \int \left ( \int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) \mathop {\mathrm {d}y} \\ f(y) &= \left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) y+ c_{1} \\ \end{align*} Where \(c_{1}\) is constant of integration. Substituting result found above for \(f(y)\) into equation (3) gives \(\phi \) \[ \phi = \int _{}^{t}\frac {\left (-\textit {\_a} \sqrt {\textit {\_a}^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) y+ c_{1} \] But since \(\phi \) itself is a constant function, then let \(\phi =c_{2}\) where \(c_{2}\) is new constant and combining \(c_{1}\) and \(c_{2}\) constants into new constant \(c_{1}\) gives the solution as \[ c_{1} = \int _{}^{t}\frac {\left (-\textit {\_a} \sqrt {\textit {\_a}^{3}-3}-y \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) y \]
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{t}\frac {\left (-\textit {\_a} \sqrt {\textit {\_a}^{3}-3}-y\right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) y &= c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{t}\frac {\left (-\textit {\_a} \sqrt {\textit {\_a}^{3}-3}-y\right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +\left (\int _{}^{t}\frac {{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}\, \sqrt {\frac {3^{\frac {1}{3}}-\textit {\_a}}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i \textit {\_a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i \textit {\_a}}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {\textit {\_a}^{3}-3}}}}{\sqrt {\textit {\_a}^{3}-3}}d \textit {\_a} +{\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{i 3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+i 3^{\frac {1}{3}}+2 i t}\, \operatorname {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-i 3^{\frac {1}{3}}-2 i t}}{6}, \frac {\sqrt {3}}{2}+\frac {i}{2}\right )}{\sqrt {t^{3}-3}}}\right ) y = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {y}{\sqrt {t^{3}-3}}=t \\ \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} \\ {} & {} & y^{\prime }=\frac {y}{\sqrt {t^{3}-3}}+t \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {y}{\sqrt {t^{3}-3}}=t \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (t \right ) \\ {} & {} & \mu \left (t \right ) \left (y^{\prime }-\frac {y}{\sqrt {t^{3}-3}}\right )=\mu \left (t \right ) t \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d t}\left (y \mu \left (t \right )\right ) \\ {} & {} & \mu \left (t \right ) \left (y^{\prime }-\frac {y}{\sqrt {t^{3}-3}}\right )=y^{\prime } \mu \left (t \right )+y \mu ^{\prime }\left (t \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (t \right ) \\ {} & {} & \mu ^{\prime }\left (t \right )=-\frac {\mu \left (t \right )}{\sqrt {t^{3}-3}} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (t \right )={\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}\left (y \mu \left (t \right )\right )\right )d t =\int \mu \left (t \right ) t d t +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (t \right )=\int \mu \left (t \right ) t d t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (t \right ) t d t +c_{1}}{\mu \left (t \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (t \right )={\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t} \\ {} & {} & y=\frac {\int t \,{\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t}d t +c_{1}}{{\mathrm e}^{\int -\frac {1}{\sqrt {t^{3}-3}}d t}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {\int t \,{\mathrm e}^{\frac {-\frac {2 \,\mathrm {I}}{3} 3^{\frac {5}{6}} \sqrt {\mathrm {-I} \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}}}\, \sqrt {\mathrm {I} \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \mathit {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\mathrm {-I} \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}}{3}, \sqrt {\frac {\mathrm {-I} \,3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}}}\right )}{\sqrt {t^{3}-3}}}d t +c_{1}}{{\mathrm e}^{\frac {-\frac {2 \,\mathrm {I}}{3} 3^{\frac {5}{6}} \sqrt {\mathrm {-I} \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \sqrt {\frac {t -3^{\frac {1}{3}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}}}\, \sqrt {\mathrm {I} \left (t +\frac {3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}\, \mathit {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\mathrm {-I} \left (t +\frac {3^{\frac {1}{3}}}{2}+\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}\right ) 3^{\frac {1}{6}}}}{3}, \sqrt {\frac {\mathrm {-I} \,3^{\frac {5}{6}}}{-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {\mathrm {I} \,3^{\frac {5}{6}}}{2}}}\right )}{\sqrt {t^{3}-3}}}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\left (\int t \,{\mathrm e}^{\frac {\mathrm {-I} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-\mathrm {I} \,3^{\frac {1}{3}}-2 \,\mathrm {I} t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{\mathrm {I} \,3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+\mathrm {I} \,3^{\frac {1}{3}}+2 \,\mathrm {I} t}\, \mathit {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-\mathrm {I} \,3^{\frac {1}{3}}-2 \,\mathrm {I} t}}{6}, \frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )}{\sqrt {t^{3}-3}}}d t +c_{1} \right ) {\mathrm e}^{\frac {\mathrm {I} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-\mathrm {I} \,3^{\frac {1}{3}}-2 \,\mathrm {I} t}\, \sqrt {\frac {3^{\frac {1}{3}}-t}{\mathrm {I} \,3^{\frac {5}{6}}+3 \,3^{\frac {1}{3}}}}\, \sqrt {3^{\frac {5}{6}}+\mathrm {I} \,3^{\frac {1}{3}}+2 \,\mathrm {I} t}\, \mathit {EllipticF}\left (\frac {3^{\frac {7}{12}} \sqrt {2}\, \sqrt {3^{\frac {5}{6}}-\mathrm {I} \,3^{\frac {1}{3}}-2 \,\mathrm {I} t}}{6}, \frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )}{\sqrt {t^{3}-3}}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 33
dsolve(diff(y(t),t)=y(t)/sqrt(t^3-3)+t,y(t), singsol=all)
\[ y \left (t \right ) = \left (\int t \,{\mathrm e}^{-\left (\int \frac {1}{\sqrt {t^{3}-3}}d t \right )}d t +c_{1} \right ) {\mathrm e}^{\int \frac {1}{\sqrt {t^{3}-3}}d t} \]
✓ Solution by Mathematica
Time used: 20.591 (sec). Leaf size: 110
DSolve[y'[t]==y[t]/Sqrt[t^3-3]+t,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{\frac {t \sqrt {1-\frac {t^3}{3}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {t^3}{3}\right )}{\sqrt {t^3-3}}} \left (\int _1^t\exp \left (-\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {K[1]^3}{3}\right ) K[1] \sqrt {1-\frac {K[1]^3}{3}}}{\sqrt {K[1]^3-3}}\right ) K[1]dK[1]+c_1\right ) \]