#### 2.1   ODE No. 1

$y'(x)-\frac {1}{\sqrt {\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3+\text {a4} x^4}}=0$ Mathematica : cpu = 0.424041 (sec), leaf count = 1117

$\left \{\left \{y(x)\to c_1-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}}\right )|\frac {\left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}\right ) \left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right )^2 \sqrt {\frac {\left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right ) \left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]\right )}} \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right ) \sqrt {\frac {\left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right ) \left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}{\left (x-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]\right )^2 \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )^2}}}{\sqrt {\text {a0}+x (\text {a1}+x (\text {a2}+x (\text {a3}+\text {a4} x)))} \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]-\text {Root}\left [\text {a4} \text {\#1}^4+\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,4\right ]\right )}\right \}\right \}$ Maple : cpu = 0.032 (sec), leaf count = 1089

$\left \{ y \left ( x \right ) =2\,{\frac { \left ( -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) +{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) ^{2}}{ \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \sqrt {{\it a4}\,{x}^{4}+{\it a3}\,{x}^{3}+{\it a2}\,{x}^{2}+{\it a1}\,x+{\it a0}}}\sqrt {{\frac { \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) }{ \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) }}}\sqrt {{\frac { \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=3 \right ) \right ) }{ \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=3 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) }}}\sqrt {{\frac { \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) \right ) }{ \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) }}}{\it EllipticF} \left ( \sqrt {{\frac { \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) }{ \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( x-{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) }}},\sqrt {{\frac { \left ( {\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=3 \right ) \right ) \left ( -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) +{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) }{ \left ( -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=3 \right ) +{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=1 \right ) \right ) \left ( -{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=4 \right ) +{\it RootOf} \left ( {\it a4}\,{{\it \_Z}}^{4}+{\it a3}\,{{\it \_Z}}^{3}+{\it a2}\,{{\it \_Z}}^{2}+{\it a1}\,{\it \_Z}+{\it a0},{\it index}=2 \right ) \right ) }}} \right ) }+{\it \_C1} \right \}$

Hand solution

\begin {equation} y^{\prime }-\frac {1}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}=0\tag {1} \end {equation}

To Do.