Confusing question in Differential first order

In summary: However, if a/A = b/B, then a linear substitution can be made to reduce the equation to separable variables. The doubts mentioned in the conversation are regarding the process of making the linear substitution and do not affect the overall conclusion that aB = bA is the necessary condition for the existence of a linear substitution.
  • #1
mepeednas
1
0
Q)
I have a first order ODE of the form
dy/dx = F(ax+by+c)/(Ax+By+C) ---> (a,b,c,A,B,C all non zero constants)

Under what condition, does there exist a linear substitution that reduces the equation to one in which the variables are separable?

(A) Never
(B) if aB = bA
(C) if bC = cB
(D) if cA = aC

Ans: I feel it is (B). But there are unclear doubts. My attempt goes below

if a/A != b/B
------------
I am aware that if a/A != b/B, then I can convert this non-homogeneous eqn into homogeneous
1) by eliminating the constants c and C using ah+bk+c=0 and Ah+Bk+C=0 and substituting x with X+h and y = Y+k. Then eqn reduces to the form (aX+bY)/(AX+BY).
2) further by substituting V=Y/X, I can convert the given eqn into a variable separable one.

if a/A = b/B
-----------
But if a/A = b/B(= t say) still I can write it as dy/dx = [t(Ax+By) + c]/(Ax+By+C) and then substituting U for Ax+By (a linear substitution), it again becomes of the form
(1/b)(dU/dx - A)= (tU+c)/U+C
which again is variable separable.

This is the doubt,
-----------------
I have done linear substitution in both the cases (x = X+h and y = Y+k) and have obtained the eqn in variable separable form. So aB = bA is not the only condition, it may as well be aB!=bA. Can anyone clarify?
 
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  • #2
Answer: The correct answer is (B). In order for a linear substitution to reduce the ODE to a separable form, the condition aB = bA must be satisfied. If a/A ≠ b/B, then it is not possible to make a linear substitution that will reduce the equation to one with separable variables.
 

FAQ: Confusing question in Differential first order

1. What is a differential first order equation?

A differential first order equation is a type of mathematical equation that involves the derivative of an unknown function. These equations are commonly used in physics and engineering to model real-world processes and phenomena.

2. How do you solve a differential first order equation?

The process of solving a differential first order equation involves finding the function that satisfies the equation. This is typically done by using techniques such as separation of variables, substitution, or integrating factors.

3. What is the difference between an ordinary and a partial differential first order equation?

An ordinary differential first order equation involves only one independent variable, while a partial differential first order equation involves multiple independent variables. Additionally, partial differential equations may also involve partial derivatives, while ordinary differential equations only involve ordinary derivatives.

4. What are the applications of differential first order equations?

Differential first order equations have a wide range of applications in various fields such as physics, engineering, economics, and biology. They are used to model and analyze various systems and processes, including population growth, chemical reactions, and electrical circuits.

5. What are some common techniques used to solve differential first order equations?

Some common techniques used to solve differential first order equations include separation of variables, substitution, integrating factors, and the method of undetermined coefficients. Each technique may be more suitable for certain types of equations, so it is important to understand and be familiar with all of them.

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