# Substitution Methods for 1st Order Diff. Eqns - Help for Beginners

• eightlgddj
In summary: Could you show me how to do it that way if it is the correct substitution?Yes, that's the substitution I came up with, I must have made a mistake after that point. Could you show me how to do it that way if it is the correct substitution?
eightlgddj
Hi, I'm new to the forum, and new to differential equations. I was wondering if someone could post a no-nonsence explanation of substitution methods for first order differential equations.

Thanks!

I don't know what u may be speaking about.Are u referring to a separable diff.eq.?

$$a(x)\frac{dy}{dx}+b(x)f(y)=0$$

Daniel.

Sorry if my post was confusing... I meant first order differential equations that are neither linear nor separable.

U mean something like that?

[atex] a(x)\left(\frac{dy}{dx}\right)^{k}+b(x)f(y)=c(x) [/tex]

The homegenous equation is separable.Therefore integrable.

Daniel.

I mean one like this:

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The homogenous equation is separable.

Daniel.

I guess that's what I don't understand. How is it separable? I've been working on this for an hour, and I can't separate the variables. Could you show me how to work it out?

I really want to understand this, because I have to teach the concept of substitution methods to the rest of my class in a few weeks. My book says that this equation is not separable, and that the homogenous equation is only separable via substitution. And that's all it says.

Last edited:
The homogenous equation is

$$2xy\frac{dy}{dx}=3y^{2}$$

Since $y\neq 0$

,u get

$$2x\frac{dy}{dx}=3y$$

,separate variables & integrate to get

$$y_{hom}(x)=Cx^{3/2}$$

Now,apply Lagrange's method to find the particular solution of the nonhomogenous one.

Daniel.

I'm sorry, what's Lagrange's method? Its not in my book. I don't think its supposed to be that complicated of a solution, this is chapter 1 ODE stuff. There is no way to solve this problem by substitution?

Yes it can,make the substitution

$$y^{2}(x)=u(x)$$

Daniel.

What happened to 4x^2?

It's there in the RHS,that substitution simplifies the integration of the ODE...

Daniel.

eightlgddj said:
I mean one like this:

That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.

saltydog said:
That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.

Thanks! Thats the substitution I came up with, I must have made a mistake after that point.

## 1. What is a substitution method for solving 1st order differential equations?

A substitution method is a technique used to solve differential equations by replacing the dependent variable with a new variable. This new variable is chosen in a way that simplifies the equation and makes it easier to solve.

## 2. How do I know when to use a substitution method for solving a 1st order differential equation?

A substitution method is typically used when the differential equation is not in a form that can be solved directly, such as when it is not separable or linear. It is also useful when the equation contains multiple variables or functions.

## 3. What are some common substitution methods used for solving 1st order differential equations?

Some common substitution methods include the change of variables method, the substitution of a power series, and the Laplace transform method. Each method has its own advantages and may be more suitable for certain types of differential equations.

## 4. Can I use a substitution method to solve any 1st order differential equation?

No, not all 1st order differential equations can be solved using a substitution method. Some equations may require more advanced techniques or may not have analytical solutions at all.

## 5. Are there any tips for beginners when using substitution methods to solve 1st order differential equations?

One tip is to carefully choose the substitution variable to simplify the equation as much as possible. It may also be helpful to practice with simpler equations before moving on to more complex ones. Additionally, double-checking the solution by plugging it back into the original equation can help ensure accuracy.

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