Final answer:
To solve the separable differential equation 6x - 4y√(x^2 + 1) dy/dx = 0, we can separate the variables and integrate both sides to find the solution. Applying the initial condition allows us to determine the value of the constant of integration.
Step-by-step explanation:
To solve the separable differential equation: 6x - 4y√(x^2 + 1) dy/dx = 0, we can separate the variables by moving dy/dx to one side and the other terms to the other side:
6x = 4y√(x^2 + 1) dy/dx
Next, we can divide both sides by 4y√(x^2 + 1) to isolate dy/dx:
dy/dx = 6x / (4y√(x^2 + 1))
Now, we can integrate both sides of the equation. On the left side, we integrate with respect to y and on the right side, we integrate with respect to x:
∫ 1 dy = ∫ (6x / (4y√(x^2 + 1))) dx
Simplifying the integration:
y = (3/2)ln|x^2 + 1| + C
Now, we can apply the initial condition y(0) = -7. Substituting x = 0 and y = -7 into the equation:
-7 = (3/2)ln|0^2 + 1| + C
-7 = (3/2)ln|1| + C
-7 = (3/2)(0) + C
C = -7
Therefore, the solution to the separable differential equation with the given initial condition is:
y = (3/2)ln|x^2 + 1| - 7