Determinants and Differential Equations Problem. "[Here] we explore a relationship between determinants and solutions to a differential equation. The 3 times 3 matrix consisting of solutions to a differential equation and their derivatives is called the Wronskian and, as we will see in later chapters, plays a pivotal role in the theory of differential equations."

andg17o7 2022-09-11 Answered
Determinants and Differential Equations Problem
"[Here] we explore a relationship between determinants and solutions to a differential equation. The 3 × 3 matrix consisting of solutions to a differential equation and their derivatives is called the Wronskian and, as we will see in later chapters, plays a pivotal role in the theory of differential equations."
This is the question following the description:
"Verify that y 1 ( x ) = cos ( 2 x ) , y 2 ( x ) = sin ( 2 x ) , y 3 ( x ) = e x are solutions to the differential equation: y y + 4 y 4 y = 0, and show that { { y 1 , y 2 , y 3 } , { y 1 , y 2 , y 3 } , { y 1 , y 2 , y 3 } } is nonzero on any interval."
Now I'm really just looking for an explanation on how I would show that it is nonzero on any interval. I completed the first part, I believe, by just taking up to the third derivative for each and plugging each set in to verify an identity of 0=0 at the end. I would then put the terms into the matrix which would give me:
{ { cos ( 2 x ) , sin ( 2 x ) , e x } , { 2 sin ( 2 x ) , 2 cos ( 2 x ) , e x } , { 4 cos ( 2 x ) , 4 sin ( 2 x ) , e x } }
So I've gotten this far and if anyone could point me in the right direction on where to go from here it would be greatly appreciated.
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Answers (2)

alinearjb
Answered 2022-09-12 Author has 10 answers
Step 1
Calculating the Wronskian involves taking the determinant of the matrix you've calculated. The typical formula given looks like
| a b c d e f g h i | = a | e f h i | b | d f g i | + c | d e g h |
= a e i a f h b d i + b f g + c d h c e g .
Step 2
When you calculate this for your matrix you'll get some function in x (since each entry is a function in x). This function is the Wronskian, and you simply examine it to make sure it is never zero.

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cubanwongux
Answered 2022-09-13 Author has 4 answers
Step 1
{ { cos ( 2 x ) , sin ( 2 x ) , e x } , { 2 sin ( 2 x ) , 2 cos ( 2 x ) , e x } , { 4 cos ( 2 x ) , 4 sin ( 2 x ) , e x } }
Step 2
This is a matrix. You're asked to prove there is no x which makes the determinant of this matrix zero.
det [ cos ( 2 x ) sin ( 2 x ) e x 2 sin ( 2 x ) 2 cos ( 2 x ) e x 4 cos ( 2 x ) 4 sin ( 2 x ) e x ] = 0

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