Did you know?
Did you know that when solving an initial value problem (IVP) like the one given, it is important to determine if the equation is exact? In this case, the equation is not exact. To solve it, we can use an integrating factor. Let's go through the steps:
1. Determine if the equation is exact: The equation is not exact since the partial derivative of -(sinx)y with respect to y (-sinx) is not equal to the partial derivative of xexp(cosx) with respect to x (exp(cosx)).
2. Using an integrating factor: Multiply the entire equation by the integrating factor, which is e^(∫(-sinx)dx). This simplifies the equation to e^(∫(-sinx)dx) * y' + e^(∫(-sinx)dx) * (sinx) * y = e^(∫(-sinx)dx) * x * exp(cosx).
3. Simplify the equation: The integrating factor e^(∫(-sinx)dx) can be found by integrating -sinx, which gives us e^(cosx). Thus, the equation becomes e^(cosx) * y' + e^(cosx) * (sinx) * y = e^(cosx) * x * exp(cosx).
4. Rewrite the left side of the equation: Apply the product rule to e^(cosx) * y, which gives us (e^(cosx) * y)' = e^(cosx) * y' + e^(cosx) * (sinx) * y. Therefore, we can rewrite the equation as (e^(cosx) * y)' = e^(cosx) * x * exp(cosx).
5. Integrate both sides: Integrating both sides of the equation gives us ∫(e^(cosx) * y)' dx = ∫(e^(cosx) * x * exp(cosx)) dx.
6. Solve the integrals: The integral on the left side simplifies to e^(cosx) * y, while the integral on the right side can be simplified using u-substitution. Let u = cosx, then du = -sinx dx. Therefore, the integral on the right side becomes ∫(x * exp(u)) du.
7. Continue solving: The integral on the right side can be evaluated to exp(u) * (x - 1) + C, where C is the constant of integration. Thus, we have e^(cosx) * y = exp(cosx) * (x - 1) + C.
8. Solve for y: Divide both sides of the equation by e^(cosx) to solve for y, which gives us y = (x - 1) + C * e^(-cosx).
9. Apply the initial condition: Substitute y(0) = 1 into the equation to find the value of C. In this case, we have 1 = -1 + C * e^(-1), which simplifies to 2 = C * e^(-1). Therefore, C = 2 * e.
10. Final solution: Substituting C = 2 * e back into the equation in step 8, we get the final solution: y = (x - 1) + 2e * e^(-cosx).
By following these steps, we can solve the given IVP and find the solution to the differential equation.