by using method of undetermined coefficients
d^2y/dx^2 -5 dy/dx+6y=xe^x
To solve the given differential equation using the method of undetermined coefficients, we assume a particular solution in the form of:
y_p = A(x)e^x
where A(x) is a polynomial of degree equal to the highest degree of the right-hand side. In this case, the highest degree of right-hand side is 1 (x^1), so we take A(x) as a polynomial of degree 1.
Now, let's substitute this particular solution back into the differential equation and differentiate accordingly:
First derivative of y_p:
y'_p = A'(x)e^x + A(x)e^x
Second derivative of y_p:
y''_p = A''(x)e^x + 2A'(x)e^x + A(x)e^x
Now, substitute y_p, y'_p, and y''_p into the given differential equation:
(A''(x)e^x + 2A'(x)e^x + A(x)e^x) - 5(A'(x)e^x + A(x)e^x) + 6(A(x)e^x) = xe^x
Simplifying the equation:
A''(x)e^x + (2A'(x) - 5A'(x) + 6A(x))e^x = xe^x
A''(x)e^x - 3A'(x)e^x + 6A(x)e^x = xe^x
Now, the exponential terms on both sides can be canceled out:
A''(x) - 3A'(x) + 6A(x) = x
We can solve this resulting polynomial equation to determine the function A(x). Once we have A(x), we can substitute it back into the assumed particular solution y_p = A(x)e^x to obtain the complete particular solution.
Note: The method of undetermined coefficients is applicable when the right-hand side of the differential equation is a polynomial multiplied by functions such as exponential, trigonometric, or logarithmic functions. If the right-hand side is a product of multiple such functions, then the particular solution needs to be modified accordingly.