Question

(ODE) I'm trying to solve a problem where I need to verify that the given functions form a fundamental set of solutions of the given DiffEq on the given interval, and form the general solution. Prompt below.

Answer 1

http://i.imgur.com/tbLzmgl.png

Answer 2

@ganeshie8 , is this where I just take the derivatives of the given set of solutions and plug them in to see if the equality holds true? Or, I dunno, how do I verify it otherwise?

And how do I form the general solution?

Answer 3

yes the given solutions are independent so just verifying them will do

Answer 4

(By the way, this is before the introduction of the Wronskian matrix, so we can't use that/it's not relevant)

Answer 5

attach some constants to get the general solution

Answer 6

\[y = c_1e^{-3x} + c_2e^{4x}\]

Answer 7

^That sounds insultingly easy, lol, that can't be true.

Answer 8

question doesn't ask you to prove the independence, it just wants you to to check whethery they work or not

Answer 9

if they work, then you can simply take their linear combination for the general solution as above ^

Answer 10

Nevermind, this is during the section that introduces the Wronskian matrix, and the answer is phrased in the form of a Wronskian, you need to use it.

Answer 11

Okay, setting up the Wronskian: Just need to remember how to write matrices in LaTeX in this type of "editor".

Answer 12

God, I'm feeling like going to bed, but I need to get all this stuff done before. @ganeshie8 , do you know how I setup Matrices in OS LaTeX?

Answer 13

(Brb for 2 minutes, computer time about to expire like right now, renewing)

Answer 14

try this

```
\left|
\begin{array}{}
a&b\\
c&d\\
\end{array}
\right|
```

Answer 15

that gives
\[
\left|
\begin{array}{}
a&b\\
c&d\\
\end{array}
\right|
\]

Answer 16

```
\[
W(y_1,y_2)=
\begin{vmatrix}
y_1&y_2\\
y_1'&y_2'
\end{vmatrix}
\]
```
\[
W(y_1,y_2)=
\begin{vmatrix}
y_1&y_2\\
y_1'&y_2'
\end{vmatrix}
\]

Answer 17

Alright, renewed the computer, I'm back.

Answer 18

\[W(y_{1},y_{2})=
\begin{vmatrix}
e^{-3x}&e^{4x}\\
-3e^{-3x}&4e^{4x}
\end{vmatrix} = 4e^x+3e^x\]

Answer 19

\[=7e^x\]

Answer 20

\[ 7e^{x} \neq 0, \]Solutions are linearly independent.

Answer 21

Alright, cool. New type of problem.