Question

tan^-1(dy÷dx)=x+y solve it by intregration

Answer 1

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Answer 2

dy/dx= tan(x+y)

Say, x+y = t.

Differentiate w.r.t x to get

1+ dy/dx = dt/dx

or, dy/dx= dt/dx -1

Thus, dt/dx - 1 = tant

or, dt/(1+tant) = dx

Integrate both sides to get

1/2{ln(sin(t+ π/4))}+ t/2= x + k, where k is any constant

Resubstituting the value of t we get the solution as

ln{sin(x+y+π/4)} + x + y = 2x + 2k

or, y = x -ln{sin(x+y+π/4)} + C , where C= 2k.

The required solution is thus

y = x -ln{sin(x+y+π/4)} + C.