Find the general solution of the DE. Write your solution explicitly. y'=(y^(2)+y^(2)cosx)^2
what, have you forgotten your algebra I?
y' = (y^2 + y^2 cosx)^2
y' = y^4 (1+cosx)^2
y'/y^4 = (1+cosx)^2
dy/y^4 = (1+cosx)^2 dx
-1/3 y^-3 = 1/4 (6x+8sinx+sin2x) + c
y = -∛(4/(3(6x+8sinx+sin2x) + c))
To find the general solution of the differential equation (DE) y' = (y^2 + y^2cosx)^2, we can follow these steps:
Step 1: Rewrite the equation in terms of y and y' only.
The given equation is already in this form.
Step 2: Separate the variables by moving all y-related terms to one side and x-related terms to the other side.
(y^2 + y^2cosx)^2 dy = dx
Step 3: Integrate both sides.
∫(y^2 + y^2cosx)^2 dy = ∫dx
Step 4: Evaluate the integrals on both sides.
This integral can be quite involved, but we'll outline the process. First, let's simplify the integrand:
(y^2 + y^2cosx)^2 = y^4 + 2y^3cosx + y^2cos^2x
Now, perform the integration using techniques like substitution, partial fractions, or expanding the integrand.
Step 5: Solve for y explicitly.
After integrating and evaluating the integrals, you will obtain an equation involving y. Solve this equation to express y explicitly.
The explicit solution to the given differential equation can be quite complex depending on the outcome of the integral and solving process. Therefore, providing an explicit solution here would not be feasible without knowing the specific values and constants involved in the integral result.
To get the explicit solution, you need to perform the integration in Step 4 and solve the resulting equation from Step 5, using techniques appropriate for the complexity of the integrand.