#### 2.51   ODE No. 51

$-h(x) (y(x)-f(x)) (y(x)-g(x)) \left (y(x)-\frac {a f(x)+b g(x)}{a+b}\right )-\frac {f'(x) (y(x)-g(x))}{f(x)-g(x)}-\frac {(y(x)-f(x)) g'(x)}{g(x)-f(x)}+y'(x)=0$ Mathematica : cpu = 1.54225 (sec), leaf count = 355

$\text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {\#1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {\#1} a^2-3 \text {\#1} a b-3 \text {\#1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\& ,\frac {\log \left (\frac {\frac {-2 a f(x) h(x)-a g(x) h(x)-b f(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3+3 a^2 b h(x)^3-3 a b^2 h(x)^3-2 b^3 h(x)^3\right )}{(a+b)^3}}}-\text {\#1}\right )}{-\text {\#1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\& \right ]=\int _1^x\frac {\left (\frac {(f(K[1])-g(K[1]))^3 \left (2 a^3 h(K[1])^3-2 b^3 h(K[1])^3-3 a b^2 h(K[1])^3+3 a^2 b h(K[1])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[1])}dK[1]+c_1,y(x)\right ]$ Maple : cpu = 0.151 (sec), leaf count = 237

$\left \{ y \left ( x \right ) ={\frac {1}{9\,{a}^{3}+18\,b{a}^{2}+18\,{b}^{2}a+9\,{b}^{3}} \left ( 2\, \left ( a-b \right ) \left ( a+2\,b \right ) \left ( a+b/2 \right ) \left ( f \left ( x \right ) -g \left ( x \right ) \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{ \left ( 2\,{\it \_a}\,{a}^{2}-{\it \_a}\,ab-{\it \_a}\,{b}^{2}-3\,{a}^{2}-3\,ab-3\,{b}^{2} \right ) \left ( {\it \_a}\,{a}^{2}+{\it \_a}\,ab-2\,{\it \_a}\,{b}^{2}+3\,{a}^{2}+3\,ab+3\,{b}^{2} \right ) \left ( 2\,{\it \_a}\,{a}^{2}+5\,{\it \_a}\,ab+2\,{\it \_a}\,{b}^{2}-3\,{a}^{2}-3\,ab-3\,{b}^{2} \right ) }}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( g \left ( x \right ) -f \left ( x \right ) \right ) ^{2} \left ( {a}^{2}+ab+{b}^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) +6\, \left ( {a}^{2}+ab+{b}^{2} \right ) \left ( \left ( a+b/2 \right ) f \left ( x \right ) +1/2\,g \left ( x \right ) \left ( a+2\,b \right ) \right ) \right ) } \right \}$