Find Wronskian of {e^(x)*cos(sqrt(x)), e^(x)*sin(sqrt(x))} Homework

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In summary, the conversation discusses finding the Wronskian of two functions using the formula W(f, g) = fg' - gf' and solving for it using e^(x)*cos(sqrt(x)) and e^(x)*sin(sqrt(x)). The attempt at a solution involves simplifying the expression, but after realizing a mistake, the correct answer is determined to be e^(2x)/(2x^1/2). The conversation also mentions using LaTeX for math expressions and provides a link to a helpful resource on the subject.
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Homework Statement


Find the Wronskian of {e^(x)*cos(sqrt(x)), e^(x)*sin(sqrt(x))}.

Homework Equations


W(f, g)=fg'-gf'

The Attempt at a Solution


W(f, g)=(e^(x)*cos(sqrt(x)))(e^(x)*cos(sqrt(x))*1/(2x^1/2))-(e^(x)*sin(sqrt(x)))(-e^(x)*sin(sqrt(x))*1/(2x^1/2)+e^(x)*cos(sqrt(x)))
After simplifying this, I got (e^(2x)(1-2*sqrt(x)*sin(sqrt(x))*cos(sqrt(x)))/(2x^(1/2)). But the correct answer is e^(2x)/(2x^1/2).
 
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  • #2
It would help a lot if you edited your post to use latex for your math expressions.
 
  • #3
I agree. With almost 250 posts, it is time to learn latex.
 
  • #5
Never mind, I found the mistake in my problem.
 

Related to Find Wronskian of {e^(x)*cos(sqrt(x)), e^(x)*sin(sqrt(x))} Homework

1. What is the Wronskian of a set of functions?

The Wronskian of a set of functions is a mathematical tool used to determine if the functions in the set are linearly independent or dependent. It is denoted by W(f1, f2, ..., fn) and is calculated by taking the determinant of a matrix containing the derivatives of the functions.

2. How do I find the Wronskian of a set of functions?

To find the Wronskian of a set of functions, first calculate the derivatives of each function. Then, arrange the derivatives in a matrix and take the determinant of the matrix. The resulting value is the Wronskian of the set of functions.

3. What is the relationship between the Wronskian and linear independence?

If the Wronskian of a set of functions is non-zero at a point, then the functions are linearly independent at that point. If the Wronskian is zero at a point, the functions may still be linearly independent, but further analysis is needed to determine this.

4. Can the Wronskian be used to determine if a set of functions is linearly independent on an interval?

Yes, the Wronskian can be used to determine if a set of functions is linearly independent on an interval. If the Wronskian is non-zero for all points in the interval, then the functions are linearly independent on that interval.

5. How is the Wronskian used in differential equations?

The Wronskian is used in differential equations to determine if a set of functions is a fundamental set of solutions. If the Wronskian is non-zero at a point, then the functions are a fundamental set of solutions and can be used to solve the differential equation.

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