# USING MATLAB TO SOLVE A HIGHER ORDER ODE

Here is an example of using MATLAB to solve an inhomogeneous higher order differential equation. The equation is:

```eqn = 'D4y - 2*D2y + Dy = t^3 +2*exp(t)'
```
```eqn =

D4y - 2*D2y + Dy = t^3 +2*exp(t)

```

The notation D4y means the 4th derivative of y, Dky means the kth derivative (where k is a positive integer).

I can solve this equation with the command dsolve. I'll call the solution sol. I'll supress printing it, because the answer will give too long a line, then use the command pretty to print it, which will make it fit reasonably.

```sol = dsolve(eqn);
pretty(sol)
```
```                                                    4
2    3   t            C9
C7 - 6 t + C10 exp(t) + 2 t exp(t) - 3 t  - t  - -- + C8 #1 + -- -
4            #2

/               /                           1/2          1/2         1/2
|    / t      \ |                        2 5    C7   54 5    t   87 5
| exp| - + #4 | | C7 - 24 t + 3 exp(t) - --------- + --------- + ------- +
\    \ 2      / \                            5           5          5

1/2  2      1/2  3           1/2              2      3    4      \ \
33 5    t    7 5    t    #3   7 5    exp(t)   15 t    3 t    t       | |
---------- + --------- + -- - ------------- - ----- - ---- - -- - 39 | | / #2 -
10          10       10         5           2      2     4       / /

/    / t      \                        1/2          1/2         1/2
| exp| - - #4 | #1 (120 t - 20 C7 + 4 5    C7 + 72 5    t + 96 5    +
\    \ 2      /

1/2  2      1/2  3            1/2              2      4        \
12 5    t  + 8 5    t  - #3 - 16 5    exp(t) + 60 t  + 5 t  + 240) | /
/

1/2
(20 (3 5    - 7)) - 6

where

/   /  1/2       \ \
|   | 5          | |
#1 = exp| t | ---- - 1/2 | |
\   \  2         / /

/   /  1/2       \ \
|   | 5          | |
#2 = exp| t | ---- + 1/2 | |
\   \  2         / /

1/2  4
#3 = 5    t

1/2
5    t
#4 = ------
2
```

In this case, the answer appears much too complicated. The next thing to try is simplify.

```pretty(simplify(sol))
```
```                                                                   4
2      3   t
42 t - C7 - 6 exp(t) + C10 exp(t) + 2 t exp(t) + 12 t  + 2 t  + -- +
4

/  1/2   \
| 5    t |
C8 exp| ------ |
\   2    /             C9
---------------- + ---------------------- + 72
/ t \                   /  1/2   \
exp| - |          / t \    | 5    t |
\ 2 /       exp| - | exp| ------ |
\ 2 /    \   2    /
```

I can also try simple.

```pretty(simple(sol))
```
```                                            /     1/2      \
| t (5    - 1) |           C9
42 t - C7 - 6 exp(t) + C10 exp(t) + C8 exp| ------------ | + ------------------- +
\      2       /      /     1/2      \
| t (5    + 1) |
exp| ------------ |
\      2       /

4
2      3   t
2 t exp(t) + 12 t  + 2 t  + -- + 72
4
```

This has the terms in a different order from the previous answer but isn't simpler.

Notice that the equation is fourth order and the solution depends on 4 constants, C7, C8, C9 and C10.