   # Wronskian The Wronskian of two differentiable functions f and g , denoted by W (f, g), is defined as the function given W ( f , g ) = | f g f ' g ' | . The function f and g are linearly independent when there exists at least one value of x for which W ( f , g ) ≠ 0 . In exercise 71-74, use the Wronskian to verify that the two functions are linearly independent. y 1 = e a x sin b x y 2 = e a x cos b x ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16.2, Problem 73E
Textbook Problem
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## Wronskian The Wronskian of two differentiable functions f and g, denoted by W (f, g), is defined as the function given W ( f , g ) = | f g f ' g ' | .The function f and g are linearly independent when there exists at least one value of x for which W ( f , g ) ≠ 0 . In exercise 71-74, use the Wronskian to verify that the two functions are linearly independent. y 1 = e a x sin b x y 2 = e a x cos b x

To determine

To verify:

The two function y1=eaxsinbxandy2=eaxcosbx are linearly independent by using Wronskian Method.

### Explanation of Solution

Given information:

The given function are y1=eaxsinbxandy2=eaxcosbx.

Concept Used:

The Wronskian of two differentiable functions f and g, denoted by W (f, g), is defined as the function given by the determinant W(f,g)=|fgf'g'|.

Proof:

First, find the derivative of the y1andy2 with respect to x, we get

y1'=beaxcosbx+aeaxsinbxandy2'=aeaxcosbxbeaxsinbx

Now, we have to find the determinant W(y1,y2)=|y1y2y1'y2'|.

W(y1,y2)=|eaxsinbxeaxcosbxbeaxcosbx+aeaxsinbx

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