Find the particular solution of the differential equation
y"(x) - y(x) = 1+e^x
Please help.
To find the particular solution of the given differential equation, we can use the method of undetermined coefficients. Here are the general steps to follow:
Step 1: Find the complementary solution (or the solution to the corresponding homogeneous equation). In this case, the homogeneous equation is y''(x) - y(x) = 0. To solve it, we assume y(x) = e^(rx), where r is a constant. We substitute this into the homogeneous equation to obtain the characteristic equation: r^2 - 1 = 0. Solving the characteristic equation, we find that r = 1 and r = -1. Therefore, the complementary solution is y_c(x) = c1 * e^x + c2 * e^(-x), where c1 and c2 are arbitrary constants.
Step 2: Find the particular solution by assuming a particular form based on the non-homogeneous term 1 + e^x. Since 1 is a constant and e^x is already present in the complementary solution, we assume a particular solution of the form y_p(x) = Ax + B, where A and B are constants to be determined.
Step 3: Substitute the particular form y_p(x) into the differential equation and solve for the unknown coefficients A and B. Plugging y_p(x) into the differential equation y''(x) - y(x) = 1 + e^x, we get:
(2A) - (Ax + B) = 1 + e^x
- Ax - B + 2A = 1 + e^x
Step 4: Collect like terms and solve for A and B. Comparing the coefficients on both sides of the equation, we have:
-A = 0 (from the x term)
- B + 2A = 1 (from the constant term)
From the first equation, we have A = 0. Substituting this value into the second equation, we get B = 1.
Step 5: Write the particular solution. Now that we have determined the values of A and B, we substitute them back into y_p(x) = Ax + B to get the particular solution:
y_p(x) = 0 * x + 1
y_p(x) = 1
Therefore, the particular solution that satisfies the given differential equation is y_p(x) = 1. The general solution of the differential equation is the sum of the complementary solution and the particular solution:
y(x) = y_c(x) + y_p(x)
y(x) = c1 * e^x + c2 * e^(-x) + 1
That's it!