Chapter 4, Problem 262RE

True or False? Justify your answer with a proof or a counterexample.

262. The differential equation y' = 3x²y − cos(x)y" is linear.

To determine

Whether the given statement is true or false.

Answer to Problem 262RE

The given statement is true.

Explanation of Solution

Given Information:

Statement is,

The differential equation $y^{\prime }=3x^{2}y-\cos \left(x\right)y^{″}$ is linear.

The differential equation,

$\begin{array}{l}{y}^{\prime }=3x^{2}y-\cos \left(x\right)y^{″}\\ \Rightarrow \left(\cos x\right)y^{″}-y^{\prime }+3x^{2}y=0\\ \Rightarrow y^{″}-\left(\frac{1}{\cos x}\right)y^{\prime }+\left(\frac{3x^2}{\cos x}\right)y=0\\ \Rightarrow y^{″}-\left(\sec x\right)y^{\prime }+\left(3x^2\sec x\right)y=0\end{array}$

This equation is in second order linear form. So, the given differential equation is linear. Hence, the statement is true.