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karush
Gold Member
MHB
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OK going to do #31 if others new OPs
I went over the examples but?
well we can't 6seem to start by a simple separation
I think direction fields can be derived with desmos
the LHS looks like a product... I think!MarkFL said:You want to begin by writing:
\(\displaystyle \frac{dy}{dx}=1+\frac{y}{x}+\left(\frac{y}{x}\right)^2\)
Next consider the substitution:
\(\displaystyle v=\frac{y}{x}\implies y=vx\implies \frac{dy}{dx}=x\frac{dv}{dx}+v\)
So, make the substitutions and simplify...what do you get?
$$\int \dfrac{1}{u^2 +1} du=\int \dfrac{1}{x}dx $$HallsofIvy said:We can write that as \(\displaystyle \frac{du}{u^2+ 1}= \frac{dx}{x}\) and integrate both sides.
$\arctan \dfrac{y}{x} - \ln x =c$ I think this it thanks everyone gteat help |
MarkFL said:You want to begin by writing:
\(\displaystyle \frac{dy}{dx}=1+\frac{y}{x}+\left(\frac{y}{x}\right)^2\)
Next consider the substitution:
\(\displaystyle v=\frac{y}{x}\implies y=vx\implies \frac{dy}{dx}=x\frac{dv}{dx}+v\)
So, make the substitutions and simplify...what do you get?
A first order homogeneous ODE (ordinary differential equation) is an equation that involves a single unknown function and its first derivative, where all of the terms in the equation are of the same degree. It is considered homogeneous because the equation can be rewritten in a way that all of the terms have the same units.
A homogeneous ODE is one in which all of the terms in the equation have the same degree, meaning they have the same units. This allows for the equation to be rewritten in a way that is more easily solvable.
To solve a first order homogeneous ODE, you can use the method of separation of variables or the method of integrating factors. Both methods involve manipulating the equation to separate the variables and then integrating to find the solution.
The main difference between a homogeneous and non-homogeneous ODE is the presence of a constant term. In a homogeneous ODE, there are no constant terms, while in a non-homogeneous ODE, there is at least one constant term. This makes solving a homogeneous ODE easier, as it can be rewritten in a way that all terms have the same units.
First order homogeneous ODEs have many real-world applications, including in physics, chemistry, and engineering. They are used to model various physical phenomena such as radioactive decay, population growth, and chemical reactions. They are also used in circuit analysis and in the study of fluid dynamics.