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 = 0.00713852 (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 = 1.018 (sec), leaf count = 1361

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