Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

52 CAPITOLUL 2
R: x(t) = e t + e −t
b) ¨x + 2˙x + x = 0 x(0) = 0, ˙x(0) = 1
R: x(t) = t · e −t
c) ¨x − 4 ˙x + 4x = 0 x(1) = 1, ˙x(1) = 0
d) ¨x + x = 0 x
π
2
R: x(t) = 3e 2t−2 − 2t · e 2t−2
= 1, ˙x
π
= 0
2
R: x(t) = sin t
e) ¨x + ˙x + x = 0 x(0) = 0, ˙x(0) = 1
R: x(t) = 2
3
2. Rezolvat¸i urmǎtoarele ecuat¸ii diferent¸iale :
a) ¨x + 3˙x + 2x = 1
1 + e t
√ 3 · e − 1
2 t · sin
2√
3t
3
R: x(t) = e −t · ln(1 + e t ) + e −2t · ln(1 + e t ) − e −2t · C1 + e −t · C2
b) ¨x − 6˙x + 9x = 9t2 + 6t + 2
t 3
c) ¨x + x = et
2
R: x(t) = e 3t · C1 + t · e 3t · C2 + 1
t
+ e−t
2
R: x(t) = C1 · sin t + C2 · cost + 1
4 (e2t + 1) · e−t
d) ¨x − 3˙x + 2x = 2e 2t
R: x(t) = (2te t − 2e t + C1e t + C2)e t
e) ¨x − 4 ˙x + 4x = 1 + e t + e 2t