# How Can I Solve This Differential Equation for x(y)?

• gionole
gionole
Homework Statement
Help me solve differential equation
Relevant Equations
##\frac{\dot x}{\sqrt{y(1+\dot x^2)}} = \text{const}##
I'm trying to solve the following differential:

##\frac{\dot x}{\sqrt{y(1+\dot x^2)}} = \text{const}##

##\dot x## is the derivative with respect to ##y##.

How do I solve it so that I end up with ##x(y)## solution ? You can find this here, but there're 2 problems: 1) I don't understand what ##a## is and how author solves it 2) Author solves it as moving into ##\theta##, which I don't want. I prefer to know how I solve it to get ##x(y)##.

Let ##k## be the constant. If ##\dot x = \frac{dx}{dy}##, then try using algebra to solve for ##\frac{dx}{dy}##.

Gordianus
@erobz

I ended up as well(thanks to you) with ##x = \int_{y_1}^{y_2} \frac{ydy}{\sqrt{2ay - y^2}}##. Now author moves to ##\theta##, but as I told you, I want to end up with ##x(y)## and not ##x(\theta)##. Thoughts ?

well, by using ##k##, we get ##\dot x = \sqrt{\frac{k^2y}{1-k^2y}}##.. Now, we say that ##x## is the integration of RHS, right ? but using integral calculator, I end up with huge answer. I guess, that's the downside of using ##x(y)## ? and how do I find ##k## ?

gionole said:
well, by using ##k##, we get ##\dot x = \sqrt{\frac{k^2y}{1-k^2y}}##.. Now, we say that ##x## is the integration of RHS, right ? but using integral calculator, I end up with huge answer. I guess, that's the downside of using ##x(y)## ? and how do I find ##k## ?
I got mixed up thinking ##y## was not under the root. But I'm getting something different than you.

$$x' = k \sqrt{y\left( 1+ x'^2\right) }$$

Square both sides

$$x'^2 = k^2 y\left( 1+ x'^2\right)$$

$$x'^2 = \frac{k^2 y}{1-k^2y}$$

$$x' = k \sqrt{ \frac{y}{1-k^2y}}$$

?

Never mind. I see we agree now.

gionole
Thanks very much. It makes sense and I realized solving this in terms of ##x(y)## is super complicated. All good.

erobz
##\theta## is just a dummy variable for the solution technique of the integral (trigonometric substitution). Whatever your end result is in terms of ##\theta## you would invert ##\theta(y)## via:

$$\tan \theta = \frac{ky}{\sqrt{y - k^2 y^2 } }$$

to get ##x(y)##...it's going to be quite the mess I think!

EDIT: Looking at the form in the paper it wouldn't be terrible, but ##y## is going to be inside an ##\arctan## function as well as outside of it in the ##\sin\theta ## term of the solution.

Last edited:
gionole
I'll sugguest assuame k as constant. Then take derivative of this like dx/dy.

## 1. What are the common methods to solve a differential equation for x(y)?

Common methods to solve a differential equation for x(y) include separation of variables, integrating factors, characteristic equations for linear differential equations, and numerical methods such as Euler's method or the Runge-Kutta methods. The choice of method depends on the type and complexity of the differential equation.

## 2. How do I determine if a differential equation is linear or nonlinear?

A differential equation is linear if it can be written in the form $$a_n(y) \frac{d^n x}{dy^n} + a_{n-1}(y) \frac{d^{n-1} x}{dy^{n-1}} + \ldots + a_1(y) \frac{dx}{dy} + a_0(y) x = g(y)$$, where the coefficients $$a_i(y)$$ and the function $$g(y)$$ depend only on the independent variable $$y$$ and not on the dependent variable $$x$$ or its derivatives. If the equation cannot be written in this form, it is nonlinear.

## 3. What is the separation of variables method and when can it be used?

The separation of variables method involves rewriting a differential equation so that each variable appears on a different side of the equation, allowing for the integration of both sides independently. This method can be used when the differential equation can be expressed in the form $$\frac{dx}{dy} = f(x)g(y)$$. After separating the variables, we integrate both sides to find the solution.

## 4. How do I apply the integrating factor method to solve a first-order linear differential equation?

To apply the integrating factor method to a first-order linear differential equation of the form $$\frac{dx}{dy} + P(y)x = Q(y)$$, we multiply both sides by an integrating factor $$\mu(y) = e^{\int P(y) dy}$$. This transforms the equation into $$\frac{d}{dy} [\mu(y)x] = \mu(y)Q(y)$$. We then integrate both sides with respect to $$y$$ to find the solution for $$x(y)$$.

## 5. What are numerical methods for solving differential equations, and when are they used?

Numerical methods, such as Euler's method and the Runge-Kutta methods, are used to approximate solutions of differential equations when an analytical solution is difficult or impossible to obtain. These methods involve discretizing the independent variable and iteratively solving for the dependent variable. They are particularly useful for complex or nonlinear differential equations where traditional methods do not work.

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