HELP First Order DE using Substitution Method

In summary, the Substitution Method is a technique used to solve First Order Differential Equations by replacing the dependent variable with a new variable. The steps involved are identifying the dependent variable, substituting it with a new variable, rewriting the equation, solving for the new variable, and substituting it back into the original equation. However, this method has limitations and may not work for all types of equations. It cannot be used for higher order differential equations. Tips for using the Substitution Method effectively include choosing the substitution variable carefully, checking for restrictions, and practicing regularly.
  • #1
danerape
32
0
I feel as if I have made the correct substitution, what am I missing? See Attachment.

Thanks,

Dane
 

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  • #2
Any ideas at all?
 
  • #3
danerape said:
Any ideas at all?

v=y/x looks like a better bet to me. If you divide the numerator and denominator of the RHS by x you can express it completely in terms of v. Should be easy to separate.
 
  • #4
Thanks,

Dane
 

1. What is the Substitution Method and how does it help solve First Order Differential Equations?

The Substitution Method is a technique used to solve First Order Differential Equations, which are equations that involve a function and its derivative. This method involves replacing the dependent variable with a new variable, which allows us to transform the equation into a more manageable form and solve for the new variable. This helps us find the original solution to the equation.

2. What are the steps involved in using the Substitution Method to solve a First Order Differential Equation?

The steps involved in using the Substitution Method are as follows:
1. Identify the dependent variable and its derivative in the given equation.
2. Substitute the dependent variable with a new variable, usually denoted as u.
3. Rewrite the equation in terms of the new variable.
4. Solve for the new variable using algebraic techniques.
5. Once the new variable has been solved for, substitute it back into the original equation to find the solution.

3. Are there any limitations to using the Substitution Method to solve First Order Differential Equations?

Yes, the Substitution Method can only be used to solve certain types of First Order Differential Equations. It is most effective for equations that are separable or homogeneous. In some cases, it may not be possible to find a suitable substitution, making it difficult to solve the equation using this method.

4. Can the Substitution Method be used to solve higher order differential equations?

No, the Substitution Method is specifically designed to solve First Order Differential Equations. For higher order equations, other methods such as the Reduction of Order or Variation of Parameters may be used.

5. Are there any tips or tricks for using the Substitution Method effectively?

Yes, here are a few tips to keep in mind when using the Substitution Method:
- Choose the substitution variable carefully, taking into account the structure of the equation.
- Be mindful of any restrictions on the domain of the function.
- Check your solution by substituting it back into the original equation.
- Practice, practice, practice! The more you use this method, the more comfortable you will become with it.

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