2.1875   ODE No. 1875

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

$$\{f(t)(ax(t)+by(t))+x^{\prime }(t)=g(t),f(t)(cx(t)+dy(t))+y^{\prime }(t)=h(t)\}$$ ✗ Mathematica : cpu = 0.00708861 (sec), leaf count = 0 , could not solve

DSolve[{f[t]*(a*x[t] + b*y[t]) + Derivative[1][x][t] == g[t], f[t]*(c*x[t] + d*y[t]) + Derivative[1][y][t] == h[t]}, {x[t], y[t]}, t]

✓ Maple : cpu = 0.918 (sec), leaf count = 1633

$$\left\{\{x\left(t\right)=1({\mathrm{e}}^{\frac{1}{2da-2bc}\int f\left(t\right)\sqrt{-da+bc}\mathrm{d}t(\sqrt{\frac{-a^2+2da-4bc-d^2}{da-bc}}(da-bc)+(a+d)\sqrt{-da+bc})}\mathit{\_}\mathit{C}\mathit{2}\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}+{\mathrm{e}}^{-\frac{1}{2da-2bc}(\sqrt{\frac{-a^2+2da-4bc-d^2}{da-bc}}(da-bc)-(a+d)\sqrt{-da+bc})\int f\left(t\right)\sqrt{-da+bc}\mathrm{d}t}\mathit{\_}\mathit{C}\mathit{1}\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}-\int -\frac{g\left(t\right)d{\left(f\left(t\right)\right)}^2-b{\left(f\left(t\right)\right)}^{2}h\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right)g\left(t\right)}{{\left(f\left(t\right)\right)}^2}{\mathrm{e}}^{-\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}-a-d)}\mathrm{d}t{\mathrm{e}}^{\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}-a-d)}+\int -\frac{g\left(t\right)d{\left(f\left(t\right)\right)}^2-b{\left(f\left(t\right)\right)}^{2}h\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right)g\left(t\right)}{{\left(f\left(t\right)\right)}^2}{\mathrm{e}}^{\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}+a+d)}\mathrm{d}t{\mathrm{e}}^{-\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}+a+d)})\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}},y\left(t\right)=(-\frac{d}{2b}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}+\frac{1}{b}(\frac{1}{2}+\frac{a}{2}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}})){\mathrm{e}}^{\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}-a-d)}\int -\frac{g\left(t\right)d{\left(f\left(t\right)\right)}^2-b{\left(f\left(t\right)\right)}^{2}h\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right)g\left(t\right)}{{\left(f\left(t\right)\right)}^2}{\mathrm{e}}^{-\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}-a-d)}\mathrm{d}t+(\frac{d}{2b}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}+\frac{1}{b}(\frac{1}{2}-\frac{a}{2}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}})){\mathrm{e}}^{-\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}+a+d)}\int -\frac{g\left(t\right)d{\left(f\left(t\right)\right)}^2-b{\left(f\left(t\right)\right)}^{2}h\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right)g\left(t\right)}{{\left(f\left(t\right)\right)}^2}{\mathrm{e}}^{\frac{\int f\left(t\right)\mathrm{d}t}{2}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}\sqrt{-da+bc}+a+d)}\mathrm{d}t+(-\frac{d^2}{2b}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}+\frac{d}{b}(\frac{1}{2}+a\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}})-2\frac{c}{\sqrt{-da+bc}}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}+\frac{1}{b}(-\frac{a^2}{2}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}-\frac{a}{2}))\mathit{\_}\mathit{C}\mathit{2}{\mathrm{e}}^{\frac{1}{2da-2bc}\int f\left(t\right)\sqrt{-da+bc}\mathrm{d}t(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}(da-bc)+(a+d)\sqrt{-da+bc})}+(\frac{d^2}{2b}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}+\frac{d}{b}(\frac{1}{2}-a\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}})+2\frac{c}{\sqrt{-da+bc}}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}+\frac{1}{b}(\frac{a^2}{2}\frac{1}{\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}}\frac{1}{\sqrt{-da+bc}}-\frac{a}{2}))\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{1}{2da-2bc}(\sqrt{-\frac{a^2-2da+4bc+d^2}{da-bc}}(da-bc)-(a+d)\sqrt{-da+bc})\int f\left(t\right)\sqrt{-da+bc}\mathrm{d}t}+\frac{g\left(t\right)}{f\left(t\right)b}\}\right\}$$