Final answer:
To solve the given differential equation eˣy dy/dx = e⁻ʸ + e⁻⁸ˣ⁻ʸ by separation of variables, we need to separate the variables on each side of the equation and integrate.
Step-by-step explanation:
To solve the given differential equation eˣy dy/dx = e⁻ʸ + e⁻⁸ˣ⁻ʸ by separation of variables, we need to separate the variables on each side of the equation.
Starting with the left side, using logarithmic identities, we can rewrite it as ln(eˣy) dy = dx.
Next, we integrate both sides with respect to their respective variables. On the left side, we integrate ln(eˣy) dy as (1/x) * eˣy + C1, where C1 is the constant of integration. On the right side, we integrate dx as x + C2, where C2 is another constant of integration.
Finally, setting the two integrals equal to each other, we get (1/x) * eˣy + C1 = x + C2, and we can solve for y.