# Solutions to Differential Equations Exercises

### Example 1

Determine whether *y* = *e*^{x} is a solution to the d.e.

*y*' + *y*" = 2*y*.

### Example 2

Determine whether *y* = *xe*^{x} is a solution to the d.e.

*y*' = *xy*.

### Example 3

Determine whether *P* = *e*^{-t} is a solution to the d.e.

### Example 4

Determine whether *y* = *x*^{2} is a solution to the d.e.

### Example 5

Determine whether *y* = *x ^{n}* is a solution to the d.e.

*x*^{2}*y*" + *ny* = *n*^{2}*y*.

### Example 6

Show that *y* = 5*x*^{2} + 2 is a solution to the d.e.

### Example 7

Show that *y* = 5*x*^{2} is not a solution to the d.e.

### Example 8

Show that *f* (*x*) = sin x is a solution to the d.e.

*f* ^{(2)}(*x*) + *f* (*x*) = 0.

### Example 9

Show that *f* (*x*) = *e*^{x}sin *x* is a solution to the d.e.

2*f* '(*x*) – 2*f* (*x*) = *f* ''(*x*).

### Example 10

Show that *y* = ln* x* is a solution to the d.e.

*e ^{y}* =

*x*

^{2}

*y*'.