Question

Find an explicit solution of the given initial-value problem. Determine the exact interval of definition by analytical methods. eʸdx−e⁻ˣdy=0,y(0)=0

Answer 1

To find an explicit solution of the given initial-value problem, we separate the variables and integrate both sides of the equation. The explicit solution is x = -e^(-y) + 1, using the initial condition y(0) = 0.

Step-by-step explanation:

To find an explicit solution of the given initial-value problem, we need to separate the variables and integrate both sides of the equation. We have the equation e^y dx - e^(-x) dy = 0. Rearranging the equation, we get dx = e^(-y) dy. Now, we integrate both sides.

∫dx = ∫e^(-y)dy

x = -e^(-y) + C

Using the initial condition y(0) = 0, we substitute the values into the equation to find the constant of integration. y(0) = -e^(-0) + C, which simplifies to 0 = -1 + C. Therefore, C = 1.

Substituting the value of C back into the equation, we have x = -e^(-y) + 1 as the explicit solution of the initial-value problem.