# Understanding First Order ODEs and Intersection of Curves

• ka_reem13
In summary, the conversation discusses introducing the perimeter as a parameter and rearranging the equation to make y the subject. However, this only results in a bunch of quadratic curves for the solution. Part b mentions finding the second set of curves for the solution and using the tangent lines at the point of intersection to determine the gradients of the integral curves. The conversation ends with a discussion on how to show that dp/dx = p/x.
ka_reem13
Homework Statement
(a) Solve the differential equation:

[x * (dy/dx)^2] - [2y*(dy/dx)] - x = 0

How many integral curves pass through each point of the (x,y) plane (except x = 0)?
why is the solution at each point not unique

(b) The differential equation:
[(dy/dx)^2] + [f(x,y)*(dy/dx)] - 1 = 0
represents a set of curves such that two curves pass through any given point. Show that these curves intersect at right angles at the point. at f = -2y/x verify this property for the point (3,4)
Relevant Equations
differential equations
I'm aware that I can introduce the perimeter p = dy/dx
then I can rearrange my equation to make y the subject, then I can show that dp/dx = p/x. However, this only gives me a bunch of quadratic curves for my solution. However given part b I see that two curves are meant to intersect each point and I don't know where I'll get the second set of curves (solutions) from.

for part b honestly I don't even know where to start

(a) The ODE is a quadratic in $\dfrac{dy}{dx}$. How many real roots does it have?

(b) If two lines $y = m_1x + c_1$ and $y = m_2x + c_2$ intersect, then the angle between them at the intersection is given by $$\cos \theta = \frac{(1,m_1)\cdot(1,m_2)}{\|(1,m_1)\|\|(1_,m_2)\|} = \frac{1 + m_1m_2}{\sqrt{1 + m_1^2}\sqrt{1 + m_2^2}}.$$ What is $\cos \theta$ if the lines intersect at right angles? To apply this to two curves, one looks at the tangent lines at the point of intersection. What are the gradients of these tangent lines if the curves are the integral curves of this ODE?

Last edited:
PeroK
ka_reem13 said:
introduce the perimeter
parameter

ka_reem13 said:
I can show that dp/dx = p/x

ka_reem13 said:
I can show that dp/dx = p/x
haruspex said:
You can? I don’t see how.
Easy, just cancel the d's.
$$\frac{dp}{dx} = \frac{\cancel dp}{\cancel dx}$$

erobz, haruspex and SammyS

## 1. What is a first order ODE?

A first order ODE (ordinary differential equation) is a mathematical equation that relates a function of one variable to its derivative. It is called "first order" because it involves only the first derivative of the function.

## 2. How do I solve a first order ODE?

There are several methods for solving a first order ODE, including separation of variables, integrating factors, and substitution. The specific method used will depend on the form of the equation and any initial conditions given.

## 3. What is the importance of first order ODEs in science?

First order ODEs are used to model a wide range of phenomena in science and engineering, including population growth, chemical reactions, and electrical circuits. They are also fundamental in understanding more complex differential equations.

## 4. Can first order ODEs have multiple solutions?

Yes, in general, a first order ODE can have multiple solutions. However, if the equation is given with initial conditions, there will typically be a unique solution that satisfies those conditions.

## 5. Are there any real-world applications of first order ODEs?

Yes, first order ODEs have many real-world applications, such as predicting the spread of infectious diseases, modeling the growth of a population, and analyzing the dynamics of chemical reactions. They are also used in engineering to design and optimize systems.

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