find general solution of y¡±-y¡¯=squre of sinx by variation parameter

To find the general solution of the given equation y'' - y' = sin^2(x) using the method of variation of parameters, follow these steps:

1. Begin by finding the complementary solution (also known as the homogeneous solution) of the associated homogenous equation y'' - y' = 0. This can be done by assuming y = e^(kx) and solving the resulting characteristic equation k^2 - k = 0. The roots of this equation are k = 0 and k = 1, so the complementary solution is y_c(x) = c1 + c2 * e^x, where c1 and c2 are constants.

2. Next, find the particular solution (also known as the particular integral) of the non-homogenous equation y'' - y' = sin^2(x). Assume a particular solution of the form y_p(x) = u(x) * y_c(x), where u(x) is a function to be determined. In this case, y_c(x) = c1 + c2 * e^x as obtained in step 1.

So, y_p(x) = u(x) * (c1 + c2 * e^x)

3. Calculate the derivatives of y_p(x) by using the product rule:
y_p'(x) = u'(x) * (c1 + c2 * e^x) + u(x) * c2 * e^x
y_p''(x) = u''(x) * (c1 + c2 * e^x) + 2 * u'(x) * c2 * e^x + u(x) * c2 * e^x

4. Substitute these derivatives back into the original non-homogenous equation:
y_p''(x) - y_p'(x) = sin^2(x)

[u''(x) * (c1 + c2 * e^x) + 2 * u'(x) * c2 * e^x + u(x) * c2 * e^x] - [u'(x) * (c1 + c2 * e^x) + u(x) * c2 * e^x] = sin^2(x)

5. Simplify the equation above by canceling out the common terms. After simplification, the resulting equation will involve the second derivative of u(x) and sin^2(x).

6. Solve the resulting differential equation for u(x). This can be done using standard techniques such as integration or guessing an appropriate solution.

7. Once u(x) is determined, substitute it back into the particular solution y_p(x) = u(x) * (c1 + c2 * e^x) obtained in step 2.

8. The general solution of the given non-homogeneous equation y'' - y' = sin^2(x) is the sum of the complementary solution (y_c(x)) obtained in step 1 and the particular solution (y_p(x)) obtained in step 7.