1.54 problem 55

Internal problem ID [3849]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 55
Date solved : Thursday, October 17, 2024 at 04:53:58 AM
CAS classification : [_quadrature]

Solve

\begin{align*} \left (\sin \left (y\right )^{2}+x \cot \left (y\right )\right ) y^{\prime }&=0 \end{align*}

Factoring the ode gives these factors

\begin{align*} \tag{1} \sin \left (y\right )^{2}+x \cot \left (y\right ) &= 0 \\ \tag{2} y^{\prime } &= 0 \\ \end{align*}

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for \(y\) from

\begin{align*} \sin \left (y\right )^{2}+x \cot \left (y\right ) = 0 \end{align*}

Solving gives

\begin{align*} y = \arctan \left (-\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, \frac {\left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{36 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right )\\ y = \arctan \left (\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, -\frac {\left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{36 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right )\\ y = \arctan \left (-\frac {\sqrt {\frac {-i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, \frac {\left (-i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {\frac {-i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{72 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right )\\ y = \arctan \left (\frac {\sqrt {\frac {-i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, \frac {\left (i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {\frac {-i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}-12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{72 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right )\\ y = \arctan \left (-\frac {\sqrt {\frac {i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, \frac {\left (i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {\frac {i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{72 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right )\\ y = \arctan \left (\frac {\sqrt {\frac {i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{6}, -\frac {\left (i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {\frac {i \sqrt {3}\, \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 i \sqrt {3}\, x^{2}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}}}}}{72 \left (108 x^{2}+12 \sqrt {3}\, \sqrt {x^{4} \left (4 x^{2}+27\right )}\right )^{{1}/{3}} x}\right ) \end{align*}

Solving equation (2)

1.54.1 Solved as first order quadrature ode

Time used: 0.010 (sec)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {0\, dx} + c_1 \\ y &= c_1 \end{align*}

Figure 121: Slope field plot
\(y^{\prime } = 0\)

1.54.2 Solved as first order homogeneous class D2 ode

Time used: 0.038 (sec)

Applying change of variables \(y = u \left (x \right ) x\), then the ode becomes

\begin{align*} u^{\prime }\left (x \right ) x +u \left (x \right ) = 0 \end{align*}

Which is now solved In canonical form a linear first order is

\begin{align*} u^{\prime }\left (x \right ) + q(x)u \left (x \right ) &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=\frac {1}{x}\\ p(x) &=0 \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int \frac {1}{x}d x}\\ &= x \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu u &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (u x\right ) &= 0 \end{align*}

Integrating gives

\begin{align*} u x&= \int {0 \,dx} + c_1 \\ &=c_1 \end{align*}

Dividing throughout by the integrating factor \(x\) gives the final solution

\[ u \left (x \right ) = \frac {c_1}{x} \]

Converting \(u \left (x \right ) = \frac {c_1}{x}\) back to \(y\) gives

\begin{align*} y = c_1 \end{align*}

Figure 122: Slope field plot
\(y^{\prime } = 0\)

1.54.3 Solved as first order ode of type differential

Time used: 0.009 (sec)

Writing the ode as

\begin{align*} y^{\prime }&=0\tag {1} \end{align*}

Which becomes

\begin{align*} \left (1\right ) dy &= \left (0\right ) dx\tag {2} \end{align*}

But the RHS is complete differential because

\begin{align*} \left (0\right ) dx &= d\left (0\right ) \end{align*}

Hence (2) becomes

\begin{align*} \left (1\right ) dy &= d\left (0\right ) \end{align*}

Integrating gives

\begin{align*} y = c_1 \end{align*}

Figure 123: Slope field plot
\(y^{\prime } = 0\)

1.54.4 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\sin \left (y \left (x \right )\right )^{2}+x \cot \left (y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int 0d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\mathit {C1} \end {array} \]

1.54.5 Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

1.54.6 Maple dsolve solution

Solving time : 0.044 (sec)
Leaf size : 1623

dsolve((sin(y(x))^2+x*cot(y(x)))*diff(y(x),x) = 0, 
       y(x),singsol=all)
 

\begin{align*} y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, -\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\ y \left (x \right ) &= c_{1} \\ \end{align*}

1.54.7 Mathematica DSolve solution

Solving time : 0.249 (sec)
Leaf size : 1647

DSolve[{(Sin[y[x]]^2+x*Cot[y[x]])*D[y[x],x]==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\arccos \left (-\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to \arccos \left (-\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to -\arccos \left (\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to \arccos \left (\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to -\arccos \left (-\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \arccos \left (-\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to -\arccos \left (\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \arccos \left (\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to -\arccos \left (-\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \arccos \left (-\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to -\arccos \left (\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \arccos \left (\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to c_1 \\ \end{align*}