11.17   ODE No. 1929

$$\boxed{{\displaystyle \{\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)=-\frac{C\left(y\left(t\right)\right)f\left(\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}{\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}},\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}y\left(t\right)=-\frac{C\left(y\left(t\right)\right)f\left(\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}\right)\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)}{\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}}-g\}}}$$

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

Mathematica: cpu = 0.008001 (sec), leaf count = 110 $$\text{DSolve}[\{x^{″}(t)=-\frac{cy(t)x^{\prime }(t)f\left(\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}\right)}{\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}},y^{″}(t)=-\frac{cy(t)y^{\prime }(t)f\left(\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}\right)}{\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}}-g\},\{x(t),y(t)\},t]$$

Maple: cpu = 2.247 (sec), leaf count = 116 $$\{[\{y\left(t\right)=\mathit{O}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}(\mathit{\_}\mathit{a},[\{\left(\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{a}}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1(C\left(\mathit{\_}\mathit{a}\right)f\left(\sqrt{{\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2}\right)\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+g\sqrt{{\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2})\frac{1}{\sqrt{{\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2}}=0\},\{\mathit{\_}\mathit{a}=y\left(t\right),\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\},\{t=\int {\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{3},y\left(t\right)=\mathit{\_}\mathit{a}\}])\},\{x\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}+\int {\mathrm{e}}^{\int -\frac{C\left(y\left(t\right)\right)}{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}f\left(\sqrt{{\left(\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)\right)}^2}\right)\mathrm{d}t}\mathrm{d}t\mathit{\_}\mathit{C}\mathit{2}\}]\}$$