340 STEP 4. Review the Knowledge You Need to Score High
eln
∣∣y 1 / 2
x+ 1
∣∣
=eln 2
y^1 /^2
x+ 1
= 2
y^1 /^2 = 2 (x+ 1 )
y=( 2 )^2 (x+ 1 )^2
y= 4 (x+ 1 )^2.
Step 4. Verify result by differentiating:
dy
dx
= 4 ( 2 )(x+ 1 )=8(x+1).
Compare with
dy
dx
=
2 y
x+ 1
=
2
(
4 (x+ 1 )^2
)
(x+ 1 )
= 8 (x+ 1 ).
- y(t)=y 0 ekt
y 0 =750, 000
y( 22 )=(750, 000)e(^0.^03 )(^22 )
≈
⎧
⎪⎨
⎪⎩
1. 45109 E 6 ≈1, 451, 090 using
a TI-89,
1, 451, 094 using a TI-85.
- Step 1. Separate variables:
4 ey=
dy
dx
− 3 xey
4 ey+ 3 xey=
dy
dx
ey( 4 + 3 x)=
dy
dx
( 4 + 3 x)dx=
dy
ey
=e−ydy.
Step 2. Integrate both sides:
∫
( 4 + 3 x)dx=
∫
e−ydy
4 x+
3 x^2
2
=−e−y+C
Switch sides:e−y=−
3 x^2
2
− 4 x+C.
Step 3. Substitute given value:y( 0 )= 0
⇒e^0 = 0 − 0 +c⇒c=1.
Step 4. Take ln of both sides:
e−y=−
3 x^2
2
− 4 x+ 1
ln(e−y)=ln
(
−
3 x^2
2
− 4 x+ 1
)
y=−ln
(
1 − 4 x−
3 x^2
2
)
.
Step 5. Verify result by differentiating:
Enterd(−ln(1− 4 x− 3
(x−∧^2 )/2),x) and obtain
− 2 ( 3 x+ 4 )
3 x^2 + 8 x− 2
, which is equivalent
toey(4+ 3 x).
- y(t)=y 0 ekt
k= 0 .0625,y( 10 )=50, 000
50, 000=y 0 e^10 (^0.^0625 )
y 0 =
50, 000
e^0.^625
⎧
⎪⎪⎨
⎪⎪⎩
$26763.1 using a TI-89,
$26763.071426≈$26763. 07
using a TI-85.
- Setv(t)= 2 − 6 e−t=0. Using the [Zero]
function on your calculator, compute
t= 1 .09861.