# OEF ODE --- Introduction ---

This module actually contains 16 exercises on (elementary) ordinary differential equations.

### Coefficients order 2 I

The differential equation
.
has as a solution.

What are the values of and ?

Give the exact values of the constants.

### Coefficients order 2 II

The differential equation
has as a solution.

What are the values of and ?

Give the exact values of the constants.

### Coefficients order 2 III

The differential equation
.
has as a solution.

What are the values of and ?

Give the exact values of the constants.

### Given solutions II

We havea linear differential equation with constant coefficients
.

Knowing that the following two functions are solutions, determine this equation.

,

### Homogeneous order 2 IC

Find the solution of the differential equation
such that and .
Step 1.
:
.
Step 2.
:
{}.
Step 3.
2: 3: 4: where and are constants.
for all
Step 4.
The condition gives a condition on and which can be written :
Write C1 and C2 to denote respectively the constants et .
.
Step 5.
And the condition gives
Write the condition by using the notations C1 and C2 without taking into account the condition given in step 4.
.
Step 6.
Finally, these last two equations give = , = .
Give the exact values if necessary in the form of fractions.

In conclusion, for all is the desired solution.

### Homogeneous order 2 type I

Find the solution of the differential equation
such that , .

### Homogeneous order 2 type II

Find the solution of the differential equation
such that , .

### Homogeneous order 2 type III

Find the solution of the differential equation
such that , .

### Homogeneous order 2 type IV

Find the solution of the differential equation
such that , .

### Homogeneous order 2 mixed type

Find the solution of the differential equation
such that and .

### Homogeneous order 2 by steps

The goal of the exercise is to find the form of the solutions of the differential equation
.
Step 1.
:
.
Step 2.
{}.
Step 3.
2: 3: 4: where and denote constants.

### Limit of solution O2

Consider a differential equation
.
When this equation has
The non-existence of the limit means that even a limit as or - does not exist.

: for . . Choose "" to finish.

### Polynomial solution order 1

Find the polynomial solution y=f(x) of the differential equation
.

### Polynomial solution order 2

Find the polynomial solution of the differential equation
.

### Polynomial solution order 3

Find the polynomial solution of the differential equation
.

### Roots of solution O2

Consider a differential equation
.
When does this equation have a non-zero solution having ?
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