Potentially useful formulas:
Math 256
y′′ + p(t) y′ + q(t)y = g(t)
y 2 = u y1
y = c1 y1 + c2 y2 + y p
y1u′′ + (2 y1′ + py1) u′ = 0
y p = u1 y1 + u2 y2 , u1′ =
W1g
Wg
, u2′ = 2 , W =
W
W
y′′′ + p(t) y′′ + q(t) y′ + r(t)y = g(t)
y p = u1 y1 + u2 y2 + u3 y 3
W=
y1
y2
y3
y1′
y2′
y3′
dx = − cos(x) + C ,
y2
y1′
y2′
u1′ =
, W1 =
W1g
,
W
u2′ =
0 y2
y3
0 y2′
y3′
W2 g
,
W
, W2 =
1 y2′′ y3′′
∫ cos(x)
0 y2
, W1 =
1 y2′
y1 0
, W2 =
y1′ 1
y = c1 y1 + c2 y2 + c3 y3 + y p
y1′′ y2′′ y3′′
∫ sin(x)
y1
u3′ =
W3 g
,
W
y1 0
y3
y1′ 0
y3′
, W3 =
y1′′ 1 y3′′
dx = sin(x) + C , ∫ tan(x)dx = ln sec(x) + C ,
y1
y2
0
y1′
y2′
0
y1′′ y2′′ 1
∫ cot(x)
dx = − ln csc(x) + C
∫ sec(x)dx = ln sec(x) + tan( x) + C , ∫ csc(x)dx = ln csc(x) − cot(x) + C , ∫ sec(x)tan(x)dx = sec(x) + C ,
∫ csc(x)cot(x)dx = − csc(x) + C , ∫ sec (x)dx = tan( x) + C , ∫ csc (x)dx = − cot(x) + C
2
1
2
1
∫ sec (x)dx = 2 sec(x)tan( x) + 2 ln sec(x) + tan(x) + C
3
1
1
∫ csc (x)dx = − 2 csc(x)cot(x) + 2 ln csc(x) − cot(x) + C
3
1
1
1
1
∫ sin (x)dx = 2 x − 2 sin(x)cos(x) + C ,
∫ cos (x)dx = 2 x + 2 sin(x)cos(x) + C
∫x
∫x
2
∫e
sin(x) dx = sin(x) − x cos(x) + C ,
ax
sin(bx) dx =
∫ ln(x)
2
e ax (a sin(bx) − b cos(bx))
a2 + b2
dx = x ln(x) − x + C ,
cos(x) dx = cos(x) + x sin(x) + C
+C,
∫x e
ax
∫e
ax
dx =
cos(bx) dx =
e ax (a cos(bx) + b sin(bx))
1
( ax − 1) eax + C
a2
a2 + b 2
+C