Question
upstudy study bank question image url

Selve the following Difterential Equations
a.
b.
Solve the following Equations
a. If , Find and
b. If , show that

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.

Partial Derivatives:

a. For , and .
b. For , it is shown that .

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:

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

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!

Related Questions

Latest Calculus Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy