Answer:
Explanation:
To verify that the general solution satisfies the differential equation, we need to differentiate the implicit equation 3x^2 + 6y^2 = C with respect to x and check if it matches the given differential equation 3x + 6yy' = 0.
Differentiating both sides of the implicit equation with respect to x using the chain rule, we get:
6x + 12yy' = 0
Dividing both sides by 2, we get:
3x + 6yy' = 0
This matches the given differential equation, so we can conclude that the implicit equation 3x^2 + 6y^2 = C is a general solution to the differential equation 3x + 6yy' = 0.
To find the particular solution that satisfies the initial condition y = 2 when x = 5, we can plug these values into the general solution and solve for the constant C.
Substituting x = 5 and y = 2 into the general solution, we get:
3(5)^2 + 6(2)^2 = C
Simplifying the left-hand side, we get:
75 + 24 = C
C = 99
Therefore, the particular solution that satisfies the initial condition y = 2 when x = 5 is:
3x^2 + 6y^2 = 99
or, equivalently:
x^2 + 2y^2 = 33