Solve 1st Order ODE from Transcendental Equation

In summary, the conversation is about whether it is allowed to manipulate a transcendental equation to obtain a first-order differential equation for an unknown function y(x). One potential method suggested is to set f(x)=y(x)^2 and solving for f(x), but it is noted that a particular property of logarithms used in this method is not valid. The conversation ends with the realization that the resulting differential equation does not depend on another variable, which may be an issue.
  • #1
gonadas91
80
5
It is a general doubt about the following equation: Imagine I want to calculate an unknown function [tex] y(x)[/tex], and my starting equation is of the type

[tex] y(x)^{2}=\frac{1}{x^{2}Log^{2}(A(x)y(x)^{2})}[/tex]

, then, am I allowed to start with the equation

[tex] y(x)=\frac{1}{xLog(A(x)y(x)^{2})}[/tex]

and differenciate in both sides of the equation, to obtain a first order diferential equation to get y(x)?

(Note that the beginning equation is a trascendental equation, but why not trying to solve the first order ODE?)Thank you
 
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  • #2
I don't see why this wouldn't be allowed but I'd assume setting ##f(x)=y(x)^2## and solving for ##f(x)## would be simpler. You can ofc use ##log(a)^2=log(a)*log(a)=log(2a)## to remove the square on the logarithmic function.
 
  • #3
Mmm the last property of the logs is not valid. My question actually has to be with the modifications that are allowed on a differential equation. Also, the resulting differential equation using that is not depending on A(g) which annoys me somehow...
 
  • #4
oh yeah ofc it isn't haha, too late for this here
 

Related to Solve 1st Order ODE from Transcendental Equation

1. How do you solve a first order ODE from a transcendental equation?

To solve a first order ODE from a transcendental equation, you must first isolate the dependent variable on one side of the equation. Then, use integration techniques such as separation of variables or substitution to solve for the dependent variable. Finally, check for any initial conditions and apply them to the solution to get the final answer.

2. What is a transcendental equation?

A transcendental equation is an equation that contains one or more transcendental functions, such as exponential, logarithmic, or trigonometric functions. These equations cannot be solved algebraically and require numerical or approximate methods to find solutions.

3. What is a first order ODE?

A first order ODE (ordinary differential equation) is a type of differential equation that involves only one independent variable and its derivative(s). The order of an ODE refers to the highest derivative present in the equation.

4. What is the difference between a first order ODE and a second order ODE?

The main difference between a first order ODE and a second order ODE is the number of independent variables and their derivatives involved. A first order ODE involves only one independent variable and its derivative, while a second order ODE involves two independent variables and their derivatives.

5. What are some methods for solving first order ODEs from transcendental equations?

Some common methods for solving first order ODEs from transcendental equations include separation of variables, substitution, and integrating factors. Other techniques such as power series, Laplace transforms, and numerical methods can also be used for more complex equations.

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