DSolve[x'[t]+((x[t])/(t^2))==(1/(t^3)), x[t], t]

Integrate[((E^(-1/t))/(t^3)), t]

Integrate[((E^(-1/t))/(t^2)), t]

{{x[t]->1+(1/t)+(E^(1/t)) C[1]}}[[1,1,2]]

DSolve[y'[x]-((y[x])/x)==(((y[x])^2)/x), y[x], x]

DSolve[y[x](y[x]+1)+x(x-1)y'[x]==0, y[x], x]

y[x] /. DSolve[(x+2y[x]+3)+(2x+4y[x]-1)y'[x]==0, y[x], x] // Simplify

y[x] /. DSolve[(x+2y[x])+(2x+4y[x])y'[x]==0, y[x], x] // Simplify

F=(x+y) Sin[y];
{D[F, x], D[F,y]}// Simplify

Integrate[Sin[y], x]

Φ=(({{t, (t^(-1))}, {t, 3(t^(-1))}}));
t D[Φ,t]-(({{2, -1}, {3, -2}})).Φ

2Inverse[Φ]// MatrixForm

Inverse[Φ].{0, 2} // Simplify

v=((({{t, (t^(-1))}, {t, 3(t^(-1))}}))).Integrate[Inverse[Φ].{0, 2}, t] // Simplify

t D[v,t]-(({{2, -1}, {3, -2}})).v-{0, 2t} // Simplify