Question

To solve the differential equation dxdy=3ex−y with the initial condition 8y(0)=ln8, you can separate variables and integrate.

Answer 1

To solve the differential equation dy/dx = 3e^(x-y) with the initial condition 8y(0) = ln(8), you can separate variables and integrate.

Step-by-step explanation:

To solve the differential equation dy/dx = 3e^(x-y) with the initial condition 8y(0) = ln(8), you can separate variables and integrate:

First, rewrite the equation as dy = 3e^x dx - 3e^y dy

Next, rearrange the equation to separate the variables: dy + 3e^y dy = 3e^x dx

Now, integrate both sides of the equation:

∫(1 + 3e^y) dy = ∫3e^x dx

y + 3e^y = 3e^x + C (where C is the constant of integration)

Finally, substitute the initial condition 8y(0) = ln(8) to solve for C and find the specific solution.