Chapter 3.1, Problem 31P

Program Plan Intro

Program Description: Purpose of problem is to show the function $y_1=\sin x^2\text{ and }{y}_2=\cos x^2$ is linearly dependent function. Also explain the reason that there is no differential equation in the form $y^{″}+p\left(x\right)y^{\prime }+q\left(x\right)y=0$ with both p and q continuous everywhere, having both $y_1\text{ and }{y}_2$ as solution.

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.

Problem will use that if 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 .