2.477   ODE No. 477

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

$$ay(x)y^{\prime }(x{)}^2+(2x-b)y^{\prime }(x)-y(x)=0$$ ✓ Mathematica : cpu = 0.306258 (sec), leaf count = 146

$$\{\{y(x)\to -\frac{e^{\frac{c_1}{2}}\sqrt{2b+e^{c_1}-4x}}{2\sqrt{a}}\},\{y(x)\to \frac{e^{\frac{c_1}{2}}\sqrt{2b+e^{c_1}-4x}}{2\sqrt{a}}\},\{y(x)\to -\sqrt{2}{e}^{\frac{c_1}{2}}\sqrt{2a{e}^{c_1}-b+2x}\},\{y(x)\to \sqrt{2}{e}^{\frac{c_1}{2}}\sqrt{2a{e}^{c_1}-b+2x}\}\}$$

✓ Maple : cpu = 0.513 (sec), leaf count = 929

$$\{{\int }_{\mathit{\_}\mathit{b}}^{x}2\frac{-2\mathit{\_}\mathit{a}+b+\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}{4a{\left(y\left(x\right)\right)}^2+\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}+b^2-4\mathit{\_}\mathit{a}b+4{\mathit{\_}\mathit{a}}^2}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}-4\frac{a\mathit{\_}\mathit{f}}{4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+b^2-4bx+4x^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+b^2-4bx+4x^2}x+b^2-4bx+4x^2}-{\int }_{\mathit{\_}\mathit{b}}^{x}8\frac{a\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}(4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}+b^2-4\mathit{\_}\mathit{a}b+4{\mathit{\_}\mathit{a}}^2)}-2\frac{-2\mathit{\_}\mathit{a}+b+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}{{(4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}+b^2-4\mathit{\_}\mathit{a}b+4{\mathit{\_}\mathit{a}}^2)}^2}(8a\mathit{\_}\mathit{f}+4\frac{ba\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}-8\frac{\mathit{\_}\mathit{a}a\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}})\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0,{\int }_{\mathit{\_}\mathit{b}}^{x}2\frac{2\mathit{\_}\mathit{a}-b+\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}{-4a{\left(y\left(x\right)\right)}^2+\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\left(y\left(x\right)\right)}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}-b^2+4\mathit{\_}\mathit{a}b-4{\mathit{\_}\mathit{a}}^2}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}4\frac{a\mathit{\_}\mathit{f}}{-4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+b^2-4bx+4x^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+b^2-4bx+4x^2}x-b^2+4bx-4x^2}-{\int }_{\mathit{\_}\mathit{b}}^{x}8\frac{a\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}(-4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}-b^2+4\mathit{\_}\mathit{a}b-4{\mathit{\_}\mathit{a}}^2)}-2\frac{2\mathit{\_}\mathit{a}-b+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}{{(-4a{\mathit{\_}\mathit{f}}^2+\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}b-2\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}\mathit{\_}\mathit{a}-b^2+4\mathit{\_}\mathit{a}b-4{\mathit{\_}\mathit{a}}^2)}^2}(-8a\mathit{\_}\mathit{f}+4\frac{ba\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}}-8\frac{\mathit{\_}\mathit{a}a\mathit{\_}\mathit{f}}{\sqrt{4a{\mathit{\_}\mathit{f}}^2+4{\mathit{\_}\mathit{a}}^2-4\mathit{\_}\mathit{a}b+b^2}})\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=-\frac{-2x+b}{2}\frac{1}{\sqrt{-a}},y\left(x\right)=\frac{-2x+b}{2}\frac{1}{\sqrt{-a}}\}$$