Exercise 1.7
1. Solve the following differential equations.
(a)
(b)
(c)
2. Solve the following differential equations.
(a)
(b)
3. Solve the following differential equations.
(a)
, Note that
is a solution
(b)
Answer
1.
(a) Rewrite into the standard form.
This is Bernoulli's equation. So, multiply
to the both sides.
Now we let
. Then
implies that
This is linear in
. Thus rewrite this into the standard form in
. Then
Then we find
. Then
る. Multiply
to the standard form. Then the left-hand side is the derivative of the product of
and the dependent variable
.
Integrate both sides by
. Then
Thus,
Substitute
, we have
(b) Rewrite into the standard form.
This is Bernoulli's equation. Now multiply
to the both sides.
Now let
. Then
and
This is linear in
. Write this into the standard form in
.
Now we find
. Then
. Multiply
to the standard form. The left-hand side is the derivative of the product of
and
. Then
Integrate both sides by
.
Therefore,
and
(c) Rewrite this into the standard form.
This is Bernoulli's equation. Then multiply
and simplify
Now let
. Then we have
and
This is a linear differential equation in
. Now write inot the standard form in
.
Now we find the integrating factor
. Note that
. Multiply
to the standard form. Then the left-hand side is the derivative of the product of
and
.
Integrate both sides by
. We have
Therefore,
and
2.
(a) Rewrite into the standard form.
This is Bernoulli's equataion. Now multiply
to both sides.
Now let
. Then
and
This is linear in
. So, write into the standard form in
.
Now we find the integrating factor
. Note that
. Then multiply
to the standard form,The left-hand side is the derivative of the product of
and
.
Integrate both sides by
.
Therefore,
(b) In
,we let
. Then
. Now put this back into the original equation.
This is linear in
. So, write this into the standard form in
.
Now we find
. Then
. Then multiply this to the standard form. Then the left-hand side is the derivative of the product of
and
.
Integrate both sides by
.
Thereforem
and
3.
(a) Rewrite this into the standard form.
This is Riccati's equation. Since
is a solution of the above differential equation, let
. Then
. Write this into the standard form.
Simplifing, we have
. Therefore,
Here, since
, we have
(b) Rewrite this into the standard form
This is Riccati's equation. Then we find a solution of this equation. Note that
is a solution of this equation. We let
. Then
. Put these back into the standard form.
Simplyfying to get
. This is a linear differential equation. So, write in the standard form.
Now finding the integrating factir
, we have
. Multiplying
to the standard form in
. Then the left-hand side is the derivative of the product of
and
. Thus,
Integrate both sides with respect to
.
Therefore,
. Hence,