2.1756   ODE No. 1756

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

$$ay(x)y^{″}(x)+b{y}^{\prime }(x{)}^2-\frac{y(x)y^{\prime }(x)}{\sqrt{c^2+x^2}}=0$$ ✓ Mathematica : cpu = 0.717703 (sec), leaf count = 211

$$\left\{\{y(x)\to c_2\exp ({\int }_1^x-\frac{{(\frac{K[2]}{\sqrt{c^2+K[2{]}^2}}+1)}^{\frac{1}{2}/a}}{{(1-\frac{K[2]}{\sqrt{c^2+K[2{]}^2}})}^{\frac{1}{2}/a}{\int }_1^{K[2]}\frac{\exp \left(\frac{\frac{1}{2}\log (\frac{K[1]}{\sqrt{c^2+K[1{]}^2}}+1)-\frac{1}{2}\log (1-\frac{K[1]}{\sqrt{c^2+K[1{]}^2}})}{a}\right)(-\sqrt{c^2+K[1{]}^2}a-b\sqrt{c^2+K[1{]}^2})}{a\sqrt{c^2+K[1{]}^2}}dK[1]-c_1{(1-\frac{K[2]}{\sqrt{c^2+K[2{]}^2}})}^{\frac{1}{2}/a}}dK[2])\}\right\}$$ ✓ Maple : cpu = 2.506 (sec), leaf count = 75

$$\{y\left(x\right)={\left(\frac{a}{(a+b)(\frac{c_{1}a{2}^{\frac{1}{a}}{x}^{\frac{1}{a}+1}\mathrm{hypergeom}([-\frac{1}{2a},-\frac{1}{2a}-\frac{1}{2}],[-\frac{1}{a}+1],-\frac{c^2}{x^2})}{a+1}+c_2)}\right)}^{-\frac{a}{a+b}}\}$$