Question

Solve the initial value problem.

√(y) d x+(x-3) d y=0, y(4)=49
The solution is (Type an implicit solution. Type an equation using x and y as the variables.

Answer 1

To solve the initial value problem √(y) dx + (x-3) dy = 0 with the initial condition y(4) = 49, we can separate the variables and integrate to find the implicit solution -ln|3-x| = 2√(y) - 14.

Step-by-step explanation:

To solve the initial value problem √(y) dx + (x-3) dy = 0 with the initial condition y(4) = 49, we can start by rewriting the equation as √(y) dx = (3-x) dy. Now, let's separate the variables by dividing both sides of the equation by √(y)(3-x), which gives us dx/(3-x) = dy/√(y).

Integrating both sides, we have ∫dx/(3-x) = ∫dy/√(y). The integrals can be evaluated as -ln|3-x| = 2√(y) + C, where C is the constant of integration.

To find the value of C, we substitute the initial condition y(4) = 49 into the equation. This gives us -ln|3-4| = 2√(49) + C, which simplifies to ln|1| = 2(7) + C. Solving for C, we get C = -14.

Substituting C back into the equation, we have -ln|3-x| = 2√(y) - 14.