dae = {x'[t] ==y[t], x[t] +y[t] ==1};

eqns = {x'[t] - y[t]==0, x[t] + y[t] == 0};

sol = DSolve[eqns, {x,y}, t]

eqns/.sol//Simplify

eqns = {x'[t] -y[t]==Sin[t], x[t]+y[t] == 1};

sol={x[t], y[t]}/.DSolve[eqns, {x[t],y[t]},t][[1]]//Expand

eqns = {x'[t] -y[t]==Sin[t], x[t]+y[t] == 1, x[Pi] == 1/2};

sol=DSolve[eqns, {x,y}, t]

Plot[{x[t]/.sol, y[t]/.sol,(x[t]+y[t])/.sol}, {t,3,8}]

Clear[x,y,z,f,g,h,t]

eqns = {x[t] +y'[t]==f[t], 2y[t]+z'[t]==g[t], 5z[t]==h[t]};

sol = DSolve[eqns, {x,y,z}, t]

eqns/.sol//Simplify

Clear[x1,x2,x3,t,eqns]

eqns ={x2[t]+2 x3[t]+x1'[t]-2 x3'[t]==0,-27 x1[t]-22 x2[t]-17 x3[t]-x1'[t]+2 x3'[t]==0,18 x1[t]+14 x2[t]+10 x3[t]+2 x1'[t]+3 x2'[t]+2 x3'[t]==0};

A=D[eqns[[All,1]],{{x1'[t],x2'[t],x3'[t]}}]

Det[A]

sol = DSolve[eqns, {x1,x2,x3}, t]

eqns = {x''[t] == y[t], x[t] + 4y[t] == 0};

sol = DSolve[eqns, {x,y},t]

eqns/.sol//Simplify

eqns = {x''[t] == y[t], x[t] +4 y[t] == 6 Sin[t]};

sol = DSolve[eqns, {x,y},t]

eqns/.sol//Simplify

eqns = {x''[t] == y[t], x[t] + 4y[t] == Sin[t], x[Pi] == 1, x'[Pi] == 0};

sol = DSolve[eqns, {x,y},t]

Plot[{x[t]/.sol, y[t]/.sol}, {t,-7,7}]

Clear [p,q,r]

eqns = {p'''[t] - q[t] + r[t] - Sin[t], p''[t] - r[t] -Cos[t], p'[t] - q[t]+ 4, p[0] - 1, p'[0] - 1, p''[0] - 1};

Timing[sol=DSolve[(#1==0&)/@eqns,{p,q,r},t]]

eqns/.sol//Simplify