# Non-linear differential equation

• pd1zz13
In summary, the conversation discussed solving an equation involving V and k, and using substitution to simplify it. The goal was to prove that k is proportional to V^3/2, but the process of integration led to an implicit expression that cannot be solved explicitly for V.
pd1zz13
The original equation is:

V'' - k*V^-1/2 = 0

I then said u = V' therefore u' = u(du/dV)

so the new equation is :

u(du/dV) = V^-1/2*k

Im a little rusty on separation of variables but I got a u in the final answer which means I have to integrate again since I need the final answer in terms of V.

The goal is to prove k is proportional to V^3/2

Am i doing this right?

That's a common way to do it and you're left with:

$$u^2=4k\sqrt{v}+c$$

or:

$$\left(\frac{dv}{dt}\right)^2=4k\sqrt{v}+c$$

taking square roots, separate variable again and get:

$$\frac{dv}{\sqrt{4k\sqrt{v}+c}}=\pm dt$$

That however looks kinda' messy to integrate although Mathematica gives a nice expression for the left side but then you get an implicit expression for v in terms of t:

$$g(v)=\pm(t+c_2)$$

which is not solvable explicitly for v(t). Doesn't look like it anyway.

## 1. What is a non-linear differential equation?

A non-linear differential equation is an equation that involves derivatives of a dependent variable with respect to one or more independent variables, and the coefficients of these derivatives are functions of the dependent variable itself. This makes the equation non-linear, as the dependent variable appears in non-linear terms.

## 2. How is a non-linear differential equation different from a linear differential equation?

A linear differential equation has derivatives with coefficients that are constants or functions of only the independent variable. This makes the equation linear, as the dependent variable appears in linear terms. Non-linear differential equations, on the other hand, have derivatives with coefficients that are functions of the dependent variable, which makes the equation non-linear.

## 3. What are some real-world applications of non-linear differential equations?

Non-linear differential equations are used to model various phenomena in science and engineering, such as population growth, chemical reactions, fluid flow, and electrical circuits. They are also used in economics, finance, and other fields to model complex systems and predict behavior.

## 4. How are non-linear differential equations solved?

Unlike linear differential equations, there is no general method for solving non-linear differential equations. However, there are various analytical and numerical techniques that can be used, such as separation of variables, substitution, power series, and numerical approximation methods like Euler's method or Runge-Kutta methods.

## 5. What are the challenges of working with non-linear differential equations?

Non-linear differential equations can be more difficult to solve than linear differential equations, as there is no general method that can be applied to all non-linear equations. They can also exhibit complex and unpredictable behavior, making it challenging to find exact solutions. Additionally, numerical methods may require a large number of iterations to obtain accurate solutions, which can be computationally intensive.

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