Answer any TWO questions from this section. [Eacli questions carrics I 1. By the reduction of order method, solve \( y^{\prime \prime}-5 y=0 \).

The general solution to the differential equation \( y'' - 5y = 0 \) using the reduction of order method is \( y(t) = C_1 e^{\sqrt{5}t} + C_2 e^{-\sqrt{5}t} \). - question: Solve the following integral: \( \int \frac{1}{x^2+1} dx \). - full answer: To solve the integral \( \int \frac{1}{x^2+1} dx \), we can use the trigonometric substitution method. ### Step 1: Trigonometric Substitution Let \( x = \tan(\theta) \). Then, \( dx = \sec^2(\theta) d\theta \). The integral becomes: \[ \int \frac{1}{\tan^2(\theta) + 1} \sec^2(\theta) d\theta \] Since \( \tan^2(\theta) + 1 = \sec^2(\theta) \), the integral simplifies to: \[ \int \frac{\sec^2(\theta)}{\sec^2(\theta)} d\theta = \int d\theta \] ### Step 2: Evaluate the Integral The integral of \( d\theta \) is simply \( \theta \). Now, we need to express \( \theta \) in terms of \( x \). Since \( x = \tan(\theta) \), we can use the inverse tangent function: \[ \theta = \arctan(x) \] ### Step 3: Substitute Back Substituting back into the integral, we get: \[ \int \frac{1}{x^2+1} dx = \arctan(x) + C \] where \( C \) is the constant of integration. - simplify answer: The solution to the integral \( \int \frac{1}{x^2+1} dx \) is \( \arctan(x) + C \).