Final answer:
To find the homogeneous equation, replace x with 0 in the given nonhomogeneous equation. To find a particular solution of the nonhomogeneous equation, use the method of reduction of order by assuming y(x) = u(x)v(x), where u(x) is an unknown function and v(x) is a known function that satisfies the homogeneous equation. Solve for u(x) and v(x) to find the particular solution.
Step-by-step explanation:
The given nonhomogeneous equation is y'' - 7y' + 6y = x. To find the homogeneous equation, we replace x with 0 in the nonhomogeneous equation. This gives us the equation y'' - 7y' + 6y = 0. To find a particular solution y(x) of the nonhomogeneous equation, we can use the method of reduction of order. Let's assume y(x) = u(x)v(x), where u(x) is an unknown function and v(x) is the known function y1(x) that satisfies the homogeneous equation.
Plugging in y(x) = u(x)v(x) into the nonhomogeneous equation, we get (u''v + 2u'v' + uv'') - 7(u'v + uv') + 6uv = x. We simplify this equation and separate the variables to get (v''/v)u + 2(v'/v)u' + u'' - 7(u'/u - v'/v)u + 6u = x. Since the terms involving u and its derivatives are dependent on x, and the terms involving v and its derivatives are independent of x, we can equate the coefficients of x on both sides.
Solving the resulting equations, we find u(x) and v(x) which gives us the particular solution y(x) of the given nonhomogeneous equation.