Question

Compute the Wronskian of a set of functions, for x>0 {y1,y2} ={xlnx, x² lnx}
W(x)=

Answer 1

To compute the Wronskian W(x) for the set of functions { y1, y2 } = { xlnx, x²lnx }, you must first find their derivatives and then substitute them into the determinant formula. The resulting determinant will give you the Wronskian.

Step-by-step explanation:

The Wronskian of a set of functions is a determinant used in the study of differential equations to determine whether a set of solutions is linearly independent. If you are given a set of functions, { y1, y2 } = { xlnx, x²lnx } for x > 0, you can compute the Wronskian, W(x), by setting up the following determinant:

W(x) =
| y1 y2 |
| y1' y2' |

Which translates to:

W(x) =
| xlnx x²lnx |
| d/dx(xlnx) d/dx(x²lnx) |

First we calculate the derivatives:

• The derivative of y1 is y1' = lnx + 1
• The derivative of y2 is y2' = 2xlnx + x

Now, we substitute these derivatives back into the determinant:

W(x) =
| xlnx x²lnx |
| lnx + 1 2xlnx + x |

And we compute the determinant by cross multiplication:

W(x) = (xlnx)(2xlnx + x) - (x²lnx)(lnx + 1)

Simplify this expression to find the Wronskian W(x). Remember, when you compute the determinant, you multiply the main diagonal terms and subtract the product of the off-diagonal terms.