equadiff.py

Created on April 10, 2024
1.48 KB
solution ou ay'+by=0

2y'-3y=0
ce-b/ax
a=2  b=-3
y=ce3x/2

y'-2y=6x+7
y'-2y=0
a=1  b=-2
y=ce-(-2/1)x=ce2x
y=ax+b
y'=a
a-2(ax+b)=6x+7
a-2ax+a-2b=6x+7
-2a=6 a=6/-2=-3  a-2b=7
-3-2b=7  -2b=10  b=-5
y=ce(-2x)-3x-5

solution ou ay'+by=k
y'=0  y=k/b
2y'+y=1/2
a=2  b=1  k=1/2
y=k/b=1/2/1=1/2

solution generale
ce-b/ax+k/b
ce-1/2x+1/2

solution ou 
c(t)=Acos(wt+P)+Bsin(wt+P)
Acos(wt+P)=-Awsin(wt+P)
Asin(wt+P)=wAcos(wt+P)

y'+y=sin2t
g(t)=Acos(2t)+Bsin2t
g'(t)=-2Asin(2t)+sin2t(B-2A)
A+2B=0
B-2A=1
A=-2B
B-2(2-B)=1
5B=1
B=1/5
A=-2/5

solution generale
y=ce-t(-2/5)cos2t+1/5sin2t

 1.résoudre y'+10y=0
ce-a/bt a=1 b=10
ce-10t
 2.determ fontion constante
y=k/b=6/10=0.6
 3.deduire ensemble solus
y(t)=ce-10t+0.6=f(t)
 4.determ solus de lequ qui verif f(0)=0
y(0)=ce10*0+0.6
c*1+0.6
c=-0.6
f(t)=-0.6**-10t+0.6
    =0.6(1-e-10t)
 5.deduire express i(t) pour t>0
Li'+Ri=E -> 1/2i'+5i=3
enlev 1/2 -> x2
10i=6
 6.deter lim t+inf i(t)
lim i(t)=0.6 car lim e-10t =0

equa diff particuliere
u'=f(t)
y'=x2+x-1
trouver primitiv
-> x3/3+x2/2+x+c


 1.resoudr x'-4x=0
a=1 b=-4
y=ce4t
 2.determ k par g(t)=ke3t
g(t)=ke3t
g't(t)=3ke3t g'-4g=2e3t
3ke3t-4ke3t=2e3t
3k-4k=2e3t
k=-2

ay''+by'+cy=f(t)

 delta>0
f(t)=Aex1t+Bex2t
x1=(-b+sqrtdelta)/2a
x2(-b+sqrt-delta)/2a
 
 delta=0
f(t)=(A+Bt)ex0t
x=-b/2a

 delta<0
f(t)=alpha+ibeta
f(t)=(Acosbetat+Bsinbetat)ealphat
Z1=(-b-isqrtdelta)/2a
Z2=(-b-isqrt-delta)/2a

delta=b**2-4AC

z=a+ib
zbarre=a-ib
zzbarre=a2+b2

y''+w2y=0
ex
a=1  c=w2
x2+w2=0
x=iw ou -iw

f(t)=Acoswt+Bsinwt