Question
Selve the following Difterential Equations
a.
b.
Solve the following Equations
a. If
, Find
and
b. If
, show that
a.
b.
Solve the following Equations
a. If
b. If
Ask by Hamilton Rogers. in Nigeria
Dec 17,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The differential equations and partial derivatives have been solved as follows:
Differential Equations:
a.
is solved to
.
b. is integrated to find the solution.
b.
Partial Derivatives:
a. For
,
and
.
b. For , it is shown that
.
b. For
Solution
Let’s solve the differential equations and the partial derivatives step by step.
Part 1: Differential Equations
a. Solve the differential equation:
We can rewrite the equation as:
This is a separable differential equation. We can separate the variables:
Now, we can integrate both sides. The left side can be split into partial fractions:
To find
and
, we can set up the equation:
By equating coefficients, we can solve for
and
. However, for simplicity, we can directly integrate:
This gives:
Exponentiating both sides, we can express
in terms of
:
b. Solve the differential equation:
Rearranging gives:
Dividing both sides by
:
Now we can integrate both sides. The right-hand side can be simplified and integrated term by term.
Part 2: Partial Derivatives
a. If
, find
and
.
First, we find the first partial derivatives:
Now, we find the second partial derivative with respect to
:
Next, we find the mixed partial derivative:
b. If
, show that
.
First, we find the first partial derivatives:
Now, we find the second partial derivatives:
Now substituting into the equation:
Thus, we have shown that:
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The Deep Dive
To tackle the first differential equation
, we can separate the variables by rewriting the equation as
. This allows us to integrate both sides effectively. For the second equation, we can rearrange and use implicit differentiation to find the relationships between
and
that will help us express
in terms of
or vice versa.
For the partial derivatives of
, we first calculate
and then take the derivative again to find
. For the mixed derivative
, we find it’s equal to
.
When it comes to
, we differentiate
twice, starting with
and
. Calculating second derivatives leads us to show that
indeed simplifies nicely to
, verifying the equality!
Happy calculating! Keep practicing those derivatives and equations for even more fun with mathematics!