4.381   ODE No. 1381

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)=-\frac{by\left(x\right)}{x^2{(x-a)}^2}+c=0}}$$

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

Mathematica: cpu = 0.626080 (sec), leaf count = 589 $$\left\{\{y(x)\to -\frac{2c{x}^2(a-x){(1-\frac{x}{a})}^{-\frac{1}{2}\sqrt{\frac{a^2-4b}{a^2}}}(\sqrt{\frac{a^2-4b}{a^2}}{(1-\frac{x}{a})}^{\sqrt{\frac{a^2-4b}{a^2}}}{}_2{F}_1(\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}-\frac{1}{2},\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}+\frac{3}{2};\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}+\frac{5}{2};\frac{x}{a})-3{(1-\frac{x}{a})}^{\sqrt{\frac{a^2-4b}{a^2}}}{}_2{F}_1(\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}-\frac{1}{2},\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}+\frac{3}{2};\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}+\frac{5}{2};\frac{x}{a})+\sqrt{\frac{a^2-4b}{a^2}}{}_2{F}_1(-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}-\frac{1}{2},\frac{3}{2}-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}};\frac{5}{2}-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}};\frac{x}{a})+3{}_2{F}_1(-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}-\frac{1}{2},\frac{3}{2}-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}};\frac{5}{2}-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}};\frac{x}{a}))}{a(\sqrt{\frac{a^2-4b}{a^2}}-3)(\sqrt{\frac{a^2-4b}{a^2}}+3)\sqrt{\frac{a^2-4b}{a^2}}\sqrt{\frac{a-x}{a}}}+\frac{c_2(x-a{)}^{\frac{1}{2}\sqrt{\frac{a^2-4b}{a^2}}+\frac{1}{2}}{x}^{\frac{1}{2}-\frac{1}{2}\sqrt{\frac{a^2-4b}{a^2}}}}{a\sqrt{\frac{a^2-4b}{a^2}}}+c_1(x-a{)}^{\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}}{x}^{\frac{1}{2}\sqrt{1-\frac{4b}{a^2}}+\frac{1}{2}}\}\right\}$$

Maple: cpu = 0.109 (sec), leaf count = 219 $$\{y\left(x\right)=\sqrt{x(a-x)}{\left(\frac{a-x}{x}\right)}^{\frac{1}{2a}\sqrt{a^2-4b}}\mathit{\_}\mathit{C}\mathit{2}+\sqrt{x(a-x)}{\left(\frac{x}{a-x}\right)}^{\frac{1}{2a}\sqrt{a^2-4b}}\mathit{\_}\mathit{C}\mathit{1}+c\sqrt{x(a-x)}({\left(\frac{x}{a-x}\right)}^{\frac{1}{2a}\sqrt{a^2-4b}}\int \sqrt{x(a-x)}{\left(\frac{x}{a-x}\right)}^{-\frac{1}{2a}\sqrt{a^2-4b}}\mathrm{d}x-{\left(\frac{a-x}{x}\right)}^{\frac{1}{2a}\sqrt{a^2-4b}}\int \sqrt{x(a-x)}{\left(\frac{a-x}{x}\right)}^{-\frac{1}{2a}\sqrt{a^2-4b}}\mathrm{d}x)\frac{1}{\sqrt{a^2-4b}}\}$$