2.51   ODE No. 51

1. Problem in Latex
2. Mathematica input
3. Maple input

$$-h(x)(y(x)-f(x))(y(x)-g(x))(y(x)-\frac{af(x)+bg(x)}{a+b})-\frac{f^{\prime }(x)(y(x)-g(x))}{f(x)-g(x)}-\frac{(y(x)-f(x))g^{\prime }(x)}{g(x)-f(x)}+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 1.54225 (sec), leaf count = 355

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

$$\{y\left(x\right)=\frac{1}{9a^3+18b{a}^2+18b^{2}a+9b^3}(2(a-b)(a+2b)(a+b/2)(f\left(x\right)-g\left(x\right))\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{(2\mathit{\_}\mathit{a}{a}^2-\mathit{\_}\mathit{a}ab-\mathit{\_}\mathit{a}{b}^2-3a^2-3ab-3b^2)(\mathit{\_}\mathit{a}{a}^2+\mathit{\_}\mathit{a}ab-2\mathit{\_}\mathit{a}{b}^2+3a^2+3ab+3b^2)(2\mathit{\_}\mathit{a}{a}^2+5\mathit{\_}\mathit{a}ab+2\mathit{\_}\mathit{a}{b}^2-3a^2-3ab-3b^2)}d\mathit{\_}\mathit{a}+\int 1/3\frac{{(g\left(x\right)-f\left(x\right))}^2(a^2+ab+b^2)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})+6(a^2+ab+b^2)((a+b/2)f\left(x\right)+1/2g\left(x\right)(a+2b)))\}$$