Solve the following differential equation: (e ^ z - 1/(e ^ z)) * d * x - (e ^ z + 1/(e ^ z)) * d * y = 0

Question

Solve the following differential equation: 

\((\frac{e^z-1}{e^z})dx-(\frac{e^z+1}{e^z})dy=0\)

(e^z - 1/(e^z)) * d * x - (e^z + 1/(e^z)) * d * y = 0

Answer 1

\((\frac{e^2-1}{e^2})dx-(\frac{e^z+1}{e^2})dy=0\)

(1 - e^-z)dx - (1 + e^-z)dy = 0

∫(1 - e^-z)dx - ∫(1 + e^-z)dy = 0

Let z = x + iy

⇒ ∫(1 - e^{-(x + iy)})dx - ∫(1 + e^{-(x + iy)})dy = c

⇒ x + e^-(x+iy) - y + \(\frac1{i}\)e^{-(x + iy)} = c

⇒ x - y + e^{-(x + iy)}(1 + \(\frac1i\times\frac{i}i\)) = c

⇒ x - y + e^-z(1 - i) = c (\(\because\) i² = -1)

Hence, solution of given differential equation is x - y + e^-z(1 - i) = c