Differential Equations

Let
Differentiate twice and rearrange using chain rule. Eventually, you’ll reach option (c) as satisfying the condition.

The given differential equation is a non-homogeneous second-order linear equation.
The complementary solution is , and the particular solution is of the form .
Therefore, the total solution is .

The degree of a differential equation is the highest power of the highest order derivative.
In the given series expansion, the highest order derivative term is , and there is no exponent other than 1 for each derivative.
Thus, the degree is .

By solving the given differential equation using standard methods, we obtain as the solution.

Step 1: Recognize the given differential equation and convert it into the standard linear form . The given equation is: Divide the entire equation by (assuming ) to make the coefficient of equal to 1: Step 2: Identify the functions and by comparing the equation with the standard form. Step 3: Calculate the integrating factor (IF) using the formula . The integral of is . Using the logarithm property : Since : We typically take the positive value for the integrating factor, so: Step 4: Compare the result with the given options. The calculated integrating factor is , which matches option (A).