107k views
5 votes
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.

User Oleksi
by
8.5k points

1 Answer

5 votes

Final answer:

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.

User Aaron Sofaer
by
8.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories