Question

Solve the initial value problem. (4yᵉ⁴ˣʸ-5/ydy=0,y(2)=1 )The solution is (Type an equation using x and y as the variables. Type an implicit solution.)

Answer 1

To solve the provided initial value problem, we integrate both sides of the separated differential equation and then apply the given initial condition y(2) = 1 to find the constant of integration. This results in the final implicit solution: 2e⁴⁹y² = 5ln|y| + 2e⁴⁹.

Step-by-step explanation:

To solve the initial value problem (4ye⁴⁹ - 5/y)dy = 0, y(2) = 1, we first need to integrate the differential equation. We will do this by separating variables. Rewrite the equation as (4ye⁴⁹)dy = (5/y)dy and then integrate both sides with respect to y. The integral of the left side is 4e⁴⁹y²/2, and the integral of the right side is 5ln|y|. Therefore, the solution before applying the initial condition is:

2e⁴⁹y² = 5ln|y| + C

To find the constant C, we use the initial condition y(2) = 1. By substituting these values into our equation, we get:

2e⁴²(1)² = 5ln|1| + C

Since ln(1) = 0, this simplifies to:

C = 2e⁴⁹

Substitute C back into our solution to get the final implicit equation:

2e⁴⁹y² = 5ln|y| + 2e⁴⁹

This equation represents the solution to the initial value problem in implicit form.