9.13   ODE No. 1849

$$\boxed{{\displaystyle \left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)\right)\frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}y\left(x\right)-a\sqrt{b^2{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)\right)}^2+1}=0}}$$

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

Mathematica: cpu = 0.591575 (sec), leaf count = 426 $$\{\{y(x)\to \frac{\frac{(a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1){}^{3/2}}{3a{b}^2}+\frac{\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}}{a{b}^2}-\frac{c_1\log (\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}+a{b}^{2}x+b^2{c}_1)}{a}-x\log \left(b^2(\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}+a{b}^{2}x+b^2{c}_1)\right)}{2a{b}^3}+c_{3}x+c_2\},\{y(x)\to \frac{-\frac{(a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1){}^{3/2}}{3a{b}^2}-\frac{\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}}{a{b}^2}+\frac{c_1\log (\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}+a{b}^{2}x+b^2{c}_1)}{a}+x\log \left(b^2(\sqrt{a^2{b}^4{x}^2+2a{b}^4{c}_{1}x+b^4{c}_1^2-1}+a{b}^{2}x+b^2{c}_1)\right)}{2a{b}^3}+c_{3}x+c_2\}\}$$

Maple: cpu = 0.125 (sec), leaf count = 337 $$\{y\left(x\right)=\int -\frac{x}{2b}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1}-\frac{\mathit{\_}\mathit{C}\mathit{1}}{2b}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1}+\frac{1}{2b}\ln ((\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^4+a^2{b}^{4}x)\frac{1}{\sqrt{a^2{b}^4}}+\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1})\frac{1}{\sqrt{a^2{b}^4}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2}x+\mathit{\_}\mathit{C}\mathit{3},y\left(x\right)=\int \frac{x}{2b}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1}+\frac{\mathit{\_}\mathit{C}\mathit{1}}{2b}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1}-\frac{1}{2b}\ln ((\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^4+a^2{b}^{4}x)\frac{1}{\sqrt{a^2{b}^4}}+\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{b}^4+2\mathit{\_}\mathit{C}\mathit{1}{a}^2{b}^{4}x+a^2{b}^4{x}^2-1})\frac{1}{\sqrt{a^2{b}^4}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{2}x+\mathit{\_}\mathit{C}\mathit{3}\}$$