Chapter 3.1, Problem 30P

(a)

Program Plan Intro

Program Description: Purpose of problem is to show whether the function $y_1=x^3\text{ and }{y}_2=\left|x^3\right|$ is linearly dependent solution on the real line of the equation $x^2{y}^{″}-3x{y}^{\prime }+3y=0$ .

Summary introduction: Problem will use that if the quotient of function $f\left(x\right)\text{ and }g\left(x\right)$ is constant value function then the function is linearly dependent.

(b)

Program Plan Intro

Program Description: Purpose of problem is to verify that $W\left(y_1,y_2\right)$ is identically zero and also explain the reason that these facts not contradict theorem 3.

Summary introduction: Problem will use thatif suppose $y_1\text{ and }{y}_2$ are two solutions of the homogeneous second order differential equation $y^{″}+p\left(x\right)y^{\prime }+q\left(x\right)y=0$ on an open interval I on which p and q are continuous.

If $y_1\text{ and }{y}_2$ are linearly independent, then $W\left(y_1,y_2\right)=0$ on I .

If $y_1\text{ and }{y}_2$ are linearly independent, then $W\left(y_1,y_2\right)\ne 0$ at each point of I .