Answer Key for Quiz 9 (section A)
1(a) We find the equilibrium solution of dy
dx = 2yโ8 by setting the right side equal to zero, which gives
y= 4. This is a solution because it makes both sides of the differential equation equal to zero.
1(b) Separate variables and integrate:
Zdy
yโ4=Z2dx =โln |yโ4|= 2x+C,
where Cis an arbitrary constant. To find it we plug in x= 0 and y= 6, which gives
ln |6โ4|= 0 + C=โC= ln2.
Moreover, since yis above the equilibrium solution when x= 0 it has to stay there for all x, so yโ4 is always
positive and therefore we can drop the absolute values, which gives ln(yโ4) = 2x+ ln 2. Exponentiating
both sides we get
eln(yโ4) =e3x+ln 2 =e3xeln 2 =โyโ4 = 2e2x=โy= 2e2x+ 4.
2. Separate variables and integrate:
Zdy
y=Zeex
exdx
On the right we can substitute u=ex, so that du =exdx and we have
Zdy
y=Zeudu =โln |y|=eu+C=eex+C.
To find Cplug in x= 0 and y=ee, which gives
ln |ee|=ee0+C=โe=e+Cso C= 0.
Also, yhas to remain positive since it is positive at the one point we know, so we can drop the absolute
values. This gives
ln y=eex
.
Exponentiating both sides and using eln y=ywe finally get
y=eeex
.