#### 2.43   ODE No. 43

$y(x)^3 \left (4 a^2 x+3 a x^2+b\right )+y'(x)+3 x y(x)^2=0$ Mathematica : cpu = 5.87091 (sec), leaf count = 490

$\text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} J_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) J_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )}{i \sqrt {-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}+\frac {(-2 a-3 x)^2}{4 a^2}} Y_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+1}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )+\left (\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}+\frac {-2 a-3 x}{2 a}\right ) Y_{\frac {1}{2} \sqrt {\frac {4 a^3-3 b}{a^3}}}\left (-i \sqrt {\frac {(-2 a-3 x)^2}{4 a^2}-\frac {4 a^3-3 b}{4 a^3}-\frac {3}{2 a^2 y(x)}}\right )},y(x)\right ]$ Maple : cpu = 1.206 (sec), leaf count = 373

$\left \{ {\it \_C1}+{ \left ( -{{\sl K}_{{\frac {1}{2}\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}}+1}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}}\right )}\sqrt {3}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}a- \left ( a\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}-2\,a-3\,x \right ) {{\sl K}_{{\frac {1}{2}\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}}}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}}\right )} \right ) \left ( {{\sl I}_{{\frac {1}{2}\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}}+1}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}}\right )}\sqrt {3}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}a- \left ( a\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}-2\,a-3\,x \right ) {{\sl I}_{{\frac {1}{2}\sqrt {{\frac {4\,{a}^{3}-3\,b}{{a}^{3}}}}}}\left (-{\frac {\sqrt {3}}{2}\sqrt {{\frac {4\,y \left ( x \right ) {a}^{2}x+3\,a{x}^{2}y \left ( x \right ) +by \left ( x \right ) -2\,a}{{a}^{3}y \left ( x \right ) }}}}\right )} \right ) ^{-1}}=0 \right \}$