Question

If the Wronskian of f and g is tcost-sint, and if u=f+3g, v=f-g, find the Wronskian of u and v. ( I know that (f+3g)(f-g)'-(f+3g)'(f-g) but what do I do after that? Please help me.)

Answer 1

=-4(tcost-sint).just calculate as usual.

Answer 2

But how?

Answer 3

\[\left[\begin{matrix}u &v \\ u' & v'\end{matrix}\right]=\left[\begin{matrix}(f+3g) & (f-g) \\( f'+3g') & (f'-g')\end{matrix}\right]\]

Answer 4

take determinant of that matrix, you will see it cancel out and just -4(Wronskian of f and g)

Answer 5

But isn't it there's a way to do it without doing the matrix? Because I did (f+3g)(1-1)-(1+3)(f-g)=-4(f-g), is this right?

Answer 6

you are kicked out of the net? I guess!! ok, I calculate it for you

Answer 7

\[(f+3g)(f'-g')-(f'+3g')(f-g) \\= (ff'-fg'+3gf'-3gg')-(ff'-gf'+3fg'-3gg')\\=ff'-fg'+3gf'-3gg'-ff'+f'g-3fg'+3gg'\]
\[=4gf'-4fg'\\=-4(fg'-gf')\\=-4Wronskian(f,g)\\=-4(tcost-sint)\]

Answer 8

Wait a minute.

Answer 9

Thank you, you were right.

Answer 10

heehe, it's not hard for you, too. just stuck with relationship between f and g and get nowhere to go.