How to Solve a Separable Equation with Initial Condition u(0)=6?

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In summary, the given conversation discusses solving the initial value problem 4du/dt = u^2 with u(0) = 6. After multiple attempts, the solution is found to be u(t) = 12/(2-3t).
  • #1
jahrens
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4 du/dt = u^2 with initial condition u(0)=6

I have worked this multiple times, and all I get is u = (-8/(t-27))^(1/3) and it is NOT right! If anyone can help it would be very appreciated.
 
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  • #2
Okay, we are given the IVP:

\(\displaystyle 4\d{u}{t}=u^2\) where \(\displaystyle u(0)=6\)

Separating variables, switching out the dummy variables of integration and using the boundaries, we obtain:

\(\displaystyle 4\int_6^u v^{-2}\,dv=\int_0^t\,dw\)

Application of the FTOC yields:

\(\displaystyle -4\left[\frac{1}{v}\right]_6^u=[w]_0^t\)

\(\displaystyle 4\left(\frac{1}{6}-\frac{1}{u}\right)=t\)

Solving for $u$, there results:

\(\displaystyle u(t)=\frac{12}{2-3t}\)
 
  • #3
That's it! Thank you so much!
 

FAQ: How to Solve a Separable Equation with Initial Condition u(0)=6?

What is a separable equation?

A separable equation is a type of differential equation in which the dependent variable and its derivative can be separated on opposite sides of the equation. This allows for the variables to be solved independently, making it easier to find a solution.

Why are separable equations important in science?

Separable equations are important in science because they are often used to model real-world phenomena. By understanding how to solve separable equations, scientists can better understand and predict the behavior of various systems and processes.

What is the general method for solving a separable equation?

The general method for solving a separable equation is to first separate the dependent and independent variable terms on opposite sides of the equation. Then, integrate both sides with respect to their respective variables. Finally, solve for the constant of integration and substitute it back into the equation to find the solution.

What are some common applications of separable equations?

Separable equations are commonly used in physics, chemistry, biology, economics, and other fields to model a wide range of phenomena, such as population growth, chemical reactions, radioactive decay, and more. They are also used in engineering to design and analyze systems and processes.

What are some tips for solving separable equations?

Some tips for solving separable equations include carefully identifying the dependent and independent variables, being familiar with integration techniques, double checking for extraneous solutions, and practicing with a variety of examples to improve problem-solving skills.

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