#### 2.105   ODE No. 105

$a x y(x)^2+b y(x)+c x+d+x y'(x)=0$ Mathematica : cpu = 0.217699 (sec), leaf count = 473

$\left \{\left \{y(x)\to \frac {c_1 \left (i \sqrt {a} e^{-i \sqrt {a} \sqrt {c} x} \left (b \left (-\sqrt {c}\right )-i \sqrt {a} d\right ) U\left (1-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b+1,2 i \sqrt {a} \sqrt {c} x\right )-i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} U\left (-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b,2 i \sqrt {a} \sqrt {c} x\right )\right )-i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )-2 i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}-1}^b\left (2 i \sqrt {a} \sqrt {c} x\right )}{a \left (c_1 e^{-i \sqrt {a} \sqrt {c} x} U\left (-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b,2 i \sqrt {a} \sqrt {c} x\right )+e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )}\right \}\right \}$ Maple : cpu = 0.196 (sec), leaf count = 844

$\left \{ y \left ( x \right ) =-4\,{{c}^{2} \left ( -1/4\,{\it \_C1}\, \left ( {a}^{3}{c}^{2}{d}^{2}+{a}^{2}{b}^{2}{c}^{3}-2\, \left ( -ac \right ) ^{3/2}abcd-2\, \left ( -ac \right ) ^{5/2}bd \right ) {{\sl U}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( 2\,\sqrt {-ac}d+c \left ( b+2 \right ) \right ) }{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}+{c}^{2}{a}^{2} \left ( a{c}^{3} \left ( ad-b\sqrt {-ac} \right ) {{\sl M}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( 2\,\sqrt {-ac}d+c \left ( b+2 \right ) \right ) }{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}+a{c}^{3} \left ( b\sqrt {-ac}+ad \right ) {{\sl M}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+2\,adc\sqrt {-ac}+ab{c}^{2}}{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}-1/2\,{{\sl U}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+2\,adc\sqrt {-ac}+ab{c}^{2}}{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}{\it \_C1}\, \left ( bc-\sqrt {-ac}d \right ) \right ) \right ) \left ( -{\it \_C1}\, \left ( {a}^{2}{b}^{2}{c}^{4}\sqrt {-ac}+2\,ac{d}^{2} \left ( -ac \right ) ^{5/2}+{d}^{2} \left ( -ac \right ) ^{7/2} \right ) {{\sl U}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( 2\,\sqrt {-ac}d+c \left ( b+2 \right ) \right ) }{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}+4\,{c}^{4}{a}^{2} \left ( {a}^{2}{c}^{2} \left ( \sqrt {-ac}d+bc \right ) {{\sl M}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( 2\,\sqrt {-ac}d+c \left ( b+2 \right ) \right ) }{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}+{a}^{2}{c}^{2} \left ( bc-\sqrt {-ac}d \right ) {{\sl M}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+2\,adc\sqrt {-ac}+ab{c}^{2}}{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}+1/2\,{{\sl U}\left (1/2\,{\frac { \left ( -ac \right ) ^{3/2}d+2\,adc\sqrt {-ac}+ab{c}^{2}}{{c}^{2}a}},\,{\frac { \left ( -ac \right ) ^{3/2}d+ca \left ( \sqrt {-ac}d+c \left ( b+1 \right ) \right ) }{{c}^{2}a}},\,2\,x\sqrt {-ac}\right )}{\it \_C1}\, \left ( b\sqrt {-ac}+ad \right ) \right ) \right ) ^{-1}} \right \}$