1. We consider the system of differential equations x ′′ = − y − 3 (x ′ ) 2 + (y ′ ) 3 + 6 y ′′ + 2 t ; y ′′′ = y ′′ − x ′ + e x − t, with the initial conditions x( 1 ) = 2 , x ′ ( 1 ) = −4 , y( 1 ) =...

1. We consider the system of differential equations

x′′ = −y
− 3(x′)2 +
(y′)3 + 6y′′ + 2t;

y′′′ =
y′′ −
x′ +
e
x
−
t,

with the initial conditions

x(1)
= 2, x′(1)
= −4, y(1)
= −2, y′(1)
= 7 et
y′′(1)
= 6.

i) Transform the system of differential equations into an equivalent system of differential equations of order 1. Specify the number of equations of order 1 obtained.

ii) Give the initial conditions associated with the system obtained in (i).

2. Find a fundamental set of real-valued solutions to the differential equation:

y
(5)
(t)
+
y′(t)
= 0

3.
We consider the system of differential equations:

y1′
(t)
=
y1(t)
−
y2(t);

y2′(t)=y1(t)+3y2(t).

Using the elimination method, find the solution to the system that satisfies the initial conditions y1 (0) = 1 and y2 (0) = 1.