Finding a Particular Solution for a Separable Equation with Initial Condition

  • Thread starter Temp0
  • Start date
  • Tags
    Separable
In summary, the conversation discusses solving a differential equation with the initial condition of x(0)=8. The equation is rearranged and integrated, but a mistake is found in the left hand side which leads to an incorrect answer.
  • #1
Temp0
79
0

Homework Statement


dx/dt=x^2+1/25,
and find the particular solution satisfying the initial condition
x(0)=8.

Homework Equations

The Attempt at a Solution


So I began by taking out 1/25 from the right side, making the equation:
dx/dt = (1/25)(25x^2 + 1)
Then, rearranging the equation to be:
dx/(25x^2+ 1) = (1/25) dt
Taking the integral of both sides:
tan^-1(5x) = (1/25) t + c
Using the initial value x(0) = 8, I can solve for c, so first I rearrange for x.
5x = tan( (1/25)t + c)
x = tan ( (1/25)t + c)/5
Plugging in 0 and 8 into the equation gives me that c = tan^-1 (40)
However, I don't have the right answer. Can anyone help me recognize what I did wrong here? Thank you in advance.
 
Physics news on Phys.org
  • #2
dx/(25x^2+ 1) = (1/25) dt
Taking the integral of both sides:
tan^-1(5x) = (1/25) t + c

I see a mistake on the left hand side that you might spot if you do a u substitution. (u=5x, du=?)
 
  • Like
Likes Temp0
  • #3
Oh, thank you, I didn't notice that. Haha, that was careless of me.
 

1. What is a separable equation problem?

A separable equation problem is a type of mathematical problem that involves separating the variables on one side of the equation to solve for the other variable. In other words, the variables are separated into two separate terms, making it easier to solve the equation.

2. How do you solve a separable equation problem?

To solve a separable equation problem, you must first identify which variables are dependent and independent. Then, you must separate the variables onto different sides of the equation. Next, you can integrate both sides of the equation and solve for the independent variable. Finally, you can plug in the value of the independent variable to solve for the dependent variable.

3. What are the applications of separable equation problems?

Separable equation problems are commonly used in physics, engineering, and economics to model real-world situations. They can be used to solve for variables such as position, velocity, and acceleration in motion problems, as well as to analyze rates of change in economic models.

4. Can separable equation problems have more than two variables?

Yes, separable equation problems can have multiple variables as long as they can be separated on one side of the equation. However, solving for more than two variables can become more complex and may require additional techniques such as substitution or elimination.

5. Are there any common mistakes when solving separable equation problems?

One common mistake when solving separable equation problems is forgetting to include the constant of integration when integrating both sides of the equation. Another mistake is not properly separating the variables and incorrectly solving for the final answer. It is important to carefully follow the steps and check the solution to avoid making these mistakes.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
858
  • Calculus and Beyond Homework Help
Replies
8
Views
709
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
222
  • Calculus and Beyond Homework Help
Replies
11
Views
918
  • Calculus and Beyond Homework Help
Replies
2
Views
265
  • Calculus and Beyond Homework Help
Replies
1
Views
588
  • Calculus and Beyond Homework Help
Replies
6
Views
695
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
148
Back
Top