Solve the following separable equation.
y'=(4x^3-1)/y for y(1)= -3
To solve the given separable equation, we need to separate the variables and integrate both sides.
First, let's rewrite the equation in a separable form:
y' = (4x^3 - 1) / y
Next, we can multiply both sides by y to move all terms involving y to one side:
y * y' = 4x^3 - 1
Now, we can separate the variables by moving all y-related terms to one side and x-related terms to the other side:
y * y' - 4x^3 = -1
Next, we integrate both sides with respect to x:
∫(y * y' - 4x^3) dx = ∫(-1) dx
To integrate the left side, we use the product rule of integration:
∫y * y' dx - ∫4x^3 dx = ∫-1 dx
The integration of the left side can be simplified by recognizing that it is equivalent to the derivative of (y^2/2):
∫d(y^2/2) - ∫4x^3 dx = ∫-1 dx
Now, we can simplify the equation:
y^2/2 - ∫4x^3 dx = -x + C1
Integrating 4x^3 with respect to x, we get:
y^2/2 - x^4 + C2 = -x + C1
Combining the constants of integration, we can rewrite the equation as:
y^2/2 - x^4 = -x + C
To find the particular solution that satisfies y(1) = -3, we substitute x = 1 and y = -3 into the equation:
((-3)^2)/2 - 1^4 = -1 + C
9/2 - 1 = -1 + C
7/2 = -1 + C
C = 7/2 + 1
C = 9/2
Therefore, the particular solution to the given separable equation with the initial condition y(1) = -3 is:
y^2/2 - x^4 = -x + 9/2