2.412   ODE No. 412

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

$$a+x{y}^{\prime }(x{)}^2+y(x)y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.39402 (sec), leaf count = 3229

$$\{\{y(x)\to -\frac{1}{2}\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}-\frac{1}{2}\sqrt{-\frac{\frac{128}{27}{e}^{\frac{3c_1}{2}}+\frac{e^{\frac{9c_1}{2}}}{5832a^3{x}^3}}{4\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}}-\frac{32ax}{9}-\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}-\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9ax\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}+\frac{e^{3c_1}}{648a^2{x}^2}}+\frac{e^{\frac{3c_1}{2}}}{72ax}\},\{y(x)\to -\frac{1}{2}\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}+\frac{1}{2}\sqrt{-\frac{\frac{128}{27}{e}^{\frac{3c_1}{2}}+\frac{e^{\frac{9c_1}{2}}}{5832a^3{x}^3}}{4\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}}-\frac{32ax}{9}-\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}-\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9ax\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}+\frac{e^{3c_1}}{648a^2{x}^2}}+\frac{e^{\frac{3c_1}{2}}}{72ax}\},\{y(x)\to \frac{1}{2}\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}-\frac{1}{2}\sqrt{\frac{\frac{128}{27}{e}^{\frac{3c_1}{2}}+\frac{e^{\frac{9c_1}{2}}}{5832a^3{x}^3}}{4\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}}-\frac{32ax}{9}-\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}-\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9ax\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}+\frac{e^{3c_1}}{648a^2{x}^2}}+\frac{e^{\frac{3c_1}{2}}}{72ax}\},\{y(x)\to \frac{1}{2}\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}+\frac{1}{2}\sqrt{\frac{\frac{128}{27}{e}^{\frac{3c_1}{2}}+\frac{e^{\frac{9c_1}{2}}}{5832a^3{x}^3}}{4\sqrt{-\frac{16ax}{9}+\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}+\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9a\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}x}+\frac{e^{3c_1}}{1296a^2{x}^2}}}-\frac{32ax}{9}-\frac{\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}{182^{2/3}ax}-\frac{42^{2/3}(128a^4{x}^4+a{e}^{3c_1}x)}{9ax\sqrt[3]{-131072a^6{x}^6+2560a^3{e}^{3c_1}{x}^3+e^{6c_1}+\sqrt{-1073741824a^9{e}^{3c_1}{x}^9+3145728a^6{e}^{6c_1}{x}^6-3072a^3{e}^{9c_1}{x}^3+e^{12c_1}}}}+\frac{e^{3c_1}}{648a^2{x}^2}}+\frac{e^{\frac{3c_1}{2}}}{72ax}\}\}$$ ✓ Maple : cpu = 0.195 (sec), leaf count = 146

$$\{-\mathit{\_}\mathit{C}\mathit{1}{x}^2{\left(\frac{1}{x}(-y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4ax})\right)}^{\frac{3}{2}}{(-y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4ax})}^{-2}+x+\frac{4a{x}^2}{3}{(-y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4ax})}^{-2}=0,\mathit{\_}\mathit{C}\mathit{1}{x}^2{\left(\frac{1}{x}(-2y\left(x\right)-2\sqrt{{\left(y\left(x\right)\right)}^2-4ax})\right)}^{\frac{3}{2}}{(y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4ax})}^{-2}+x+\frac{4a{x}^2}{3}{(y\left(x\right)+\sqrt{{\left(y\left(x\right)\right)}^2-4ax})}^{-2}=0\}$$