9.12   ODE No. 1848

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

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

Mathematica: cpu = 594.052935 (sec), leaf count = 37 $$\text{DSolve}[y^{″}(x{)}^2(-a-3y^{\prime }(x))+y^{(3)}(x)(y^{\prime }(x{)}^2+1)=0,y(x),x]$$

Maple: cpu = 0.843 (sec), leaf count = 789 $$\{y\left(x\right)=\int \frac{\sin \left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^4+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{4}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^4{x}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^2+4{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{2}x+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{x}^2-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}{a}^3-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}{a}^{3}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{x}^2-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}a-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}ax+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2{a}^2+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2-1)\right)}{\cos \left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^4+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{4}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^4{x}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^2+4{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{2}x+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{x}^2-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}{a}^3-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}{a}^{3}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{x}^2-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}a-2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}ax+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2{a}^2+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2-1)\right)}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{3},y\left(x\right)=\int \frac{\sin \left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^4+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{4}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^4{x}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^2+4{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{2}x+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{x}^2+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}{a}^3+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}{a}^{3}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{x}^2+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}a+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}ax+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2{a}^2+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2-1)\right)}{\cos \left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^4+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{4}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^4{x}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2{a}^2+4{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}{a}^{2}x+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{a}^2{x}^2+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}{a}^3+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}{a}^{3}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{C}\mathit{2}}^2+2{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2\mathit{\_}\mathit{C}\mathit{2}x+{\mathrm{e}}^{2\mathit{\_}\mathit{Z}a}{\mathit{\_}\mathit{C}\mathit{1}}^2{x}^2+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{C}\mathit{2}a+2\cos \left(\mathit{\_}\mathit{Z}\right){\mathrm{e}}^{\mathit{\_}\mathit{Z}a}\mathit{\_}\mathit{C}\mathit{1}ax+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2{a}^2+{(\cos \left(\mathit{\_}\mathit{Z}\right))}^2-1)\right)}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{3}\}$$