Solving Differential Equation: ln |9/64| = k

In summary, the conversation discusses finding the value of k using the equation k = ln |9/64| and then attempting to use the equation for the integral of h^{-1/2} to find the value of k, which is incorrect. The correct equation for the integral of h^{-1/2} is \int x^\alpha\,dx = \frac{x^{\alpha + 1}}{\alpha + 1} + C, not \int x^{-1}\,dx = \ln |x| + C.
  • #1
kyu
12
0


i got k = ln |9/64|

then how can the next step using ln 0 doesn't make sense. what should i do?
 

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  • #2
kyu said:
i got k = ln |9/64|

then how can the next step using ln 0 doesn't make sense. what should i do?

Please show us your work.
 
  • #3
Mark44 said:
Please show us your work.

should be wrong but here goes

dh/dt = -k h^(1/2)
1/h^(1/2) = -k dt
ln 9 - ln 64 = k
ln |9/64| = k

ln 0 - ln 64 = ln |9/64| (t-0)
 
  • #4
kyu said:
should be wrong but here goes

dh/dt = -k h^(1/2)
1/h^(1/2) = -k dt
Where did the dh go?
kyu said:
ln 9 - ln 64 = k
How did you get this (above)?
kyu said:
ln |9/64| = k

ln 0 - ln 64 = ln |9/64| (t-0)
 
  • #5
kyu said:
should be wrong but here goes

dh/dt = -k h^(1/2)
1/h^(1/2) = -k dt
ln 9 - ln 64 = k
ln |9/64| = k

ln 0 - ln 64 = ln |9/64| (t-0)

The integral of [itex]h^{-1/2}[/itex] is given by the usual rule for powers, [itex]\int x^\alpha\,dx = \frac{x^{\alpha + 1}}{\alpha + 1} + C[/itex], not the exception [itex]\int x^{-1}\,dx = \ln |x| + C[/itex].
 
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1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a quantity and its rate of change. It involves one or more unknown functions and their derivatives.

2. What is the importance of differential equations?

Differential equations are important in many scientific fields, such as physics, engineering, economics, and biology. They are used to model and analyze complex systems and phenomena, and have applications in predicting future behavior and making decisions.

3. What are the types of differential equations?

The main types of differential equations are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve one independent variable, while PDEs involve multiple independent variables.

4. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some techniques include separation of variables, substitution, and using integral transforms. In some cases, a numerical solution may also be used.

5. What are some real-world applications of differential equations?

Differential equations have many applications, including modeling population growth, predicting weather patterns, analyzing electrical circuits, and understanding chemical reactions. They are also used in image processing, signal processing, and financial modeling.

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