Question

Find the solution of the differential equation dy/dx=y^2+25 that satisfies the initial condition y(2)=0.

Answer 1

To find the solution of the given differential equation, we can separate variables and integrate both sides. Applying the initial condition, we solve for the constant of integration and obtain the final solution as y = 5 tan(x-2).

Step-by-step explanation:

To find the solution of the given differential equation, we can separate variables. Rearranging the equation, we have:

1. y² = dy/dx - 25

2. Now, we can write the equation as:
   dy/(y²+25) = dx

3. Integrating both sides, we get:
   arctan(y/5) = x + C

4. Applying the initial condition y(2) = 0, we can solve for C:

   arctan(0/5) = 2 + C
   0 = 2 + C
   Therefore, C = -2

5. Substituting the value of C back into the equation, we have:
   arctan(y/5) = x - 2

6. Finally, we can solve for y by taking the tangent of both sides:
   y = 5 tan(x-2)