# Solving 1st order non-linear ODE

• MHB
• WMDhamnekar
In summary, the conversation discusses a differential equation and its solution using a trick found on the internet. The trick involves using a substitution and integrating to get an implicit solution. The conversation also mentions verifying the solution and the role of the Lambert W function in the answer provided by WolframAlpha.
WMDhamnekar
MHB
I want to solve $\d{y}{x}=\frac{3*(2x-7y)+6}{2*(2x-7y)-3}.$ I don't know its step by step solution. But using some trick of solving ordinary differential equation (which I saw on the Internet), I got the following solution:-

$-\frac{17}{21}*(3x-2y)+ln(119y-34x-48)=C$. Now how to solve this answer for y(x)?

I also want to know its step by step solution.

If any member knows the correct answer, he/she may reply with correct answer. Your Wolfram Alpha DiffEq calculator gives this answer$y(x)=\frac{21}{34}\big(W\big(-e^{(289*x)/147+c_1-1} \big)+1\big)+\frac{4*x-3}{14}$

I don't understand how to verify my answer and (1) are equivalent or not?

Last edited:
I would try the substitution:

$$\displaystyle u=2x-7y\implies \d{u}{x}=2-7\d{y}{x}\implies \d{y}{x}=\frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}$$

Our ODE then becomes:

$$\displaystyle \frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}=\frac{3u-6}{2u-3}$$

$$\displaystyle \d{u}{x}=\frac{36-17u}{2u-3}$$

$$\displaystyle \int\frac{2u-3}{36-17u}\,du=\int\,dx$$

$$\displaystyle -34u-21\ln(17u-36)+72=289x+C$$

Back-substitute for $$u$$:

$$\displaystyle -34(2x-7y)-21\ln(17(2x-7y)-36)+72=289x+C$$

I thus get the implicit solution:

$$\displaystyle \frac{17}{3}(2y-3x)-\ln(17(2x-7y)-36)=c_1$$

MarkFL said:
I would try the substitution:

$$\displaystyle u=2x-7y\implies \d{u}{x}=2-7\d{y}{x}\implies \d{y}{x}=\frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}$$

Our ODE then becomes:

$$\displaystyle \frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}=\frac{3u-6}{2u-3}$$

$$\displaystyle \d{u}{x}=\frac{36-17u}{2u-3}$$

$$\displaystyle \int\frac{2u-3}{36-17u}\,du=\int\,dx$$

$$\displaystyle -34u-21\ln(17u-36)+72=289x+C$$

Back-substitute for $$u$$:

$$\displaystyle -34(2x-7y)-21\ln(17(2x-7y)-36)+72=289x+C$$

I thus get the implicit solution:

$$\displaystyle \frac{17}{3}(2y-3x)-\ln(17(2x-7y)-36)=c_1$$
Nice! (Bow)

-Dan

MarkFL said:
I would try the substitution:

$$\displaystyle u=2x-7y\implies \d{u}{x}=2-7\d{y}{x}\implies \d{y}{x}=\frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}$$

Our ODE then becomes:

$$\displaystyle \frac{2}{7}-\frac{1}{7}\cdot\d{u}{x}=\frac{3u-6}{2u-3}$$

$$\displaystyle \d{u}{x}=\frac{36-17u}{2u-3}$$

$$\displaystyle \int\frac{2u-3}{36-17u}\,du=\int\,dx$$

$$\displaystyle -34u-21\ln(17u-36)+72=289x+C$$

Back-substitute for $$u$$:

$$\displaystyle -34(2x-7y)-21\ln(17(2x-7y)-36)+72=289x+C$$

I thus get the implicit solution:

$$\displaystyle \frac{17}{3}(2y-3x)-\ln(17(2x-7y)-36)=c_1$$

Hello,
Let u=2x-7y, $\Rightarrow\d{u}{x}=2-7\d{y}{x}\Rightarrow\d{y}{x}=\frac27-\frac17\d{u}{x}$

Our ODE then becomes $\frac27-\frac17\d{u}{x}=\frac{3u+6}{2u-3}\Rightarrow \d{u}{x}=\frac{-17u-48}{2u-3}$

$\displaystyle\int \frac{2u-3}{-17u-48}=\displaystyle\int dx$

$-34u+147*\ln{|17u+48|}=289x+C$

$-34*(2x-7y)+147*\ln{|17(2x-7y)+48|}=289x+C$$-357x+238y+147*\ln{|34x-119y+48|}=C$ is the final answer.

So It proves that there is no difference between the trick used answer and my computed step by step answer.

What is the meaning of W in the answer given by WolframAlpha DiffEq calculator?

Last edited:
I can't be certain but I suspect "W" is "Lambert's W function", also called the "omega function" or "product logarithm". It is defined as the inverse function to $$f(x)= xe^x$$. https://en.wikipedia.org/wiki/Lambert_W_function.

## 1. What is a first-order non-linear ODE?

A first-order non-linear ODE (ordinary differential equation) is a mathematical equation that involves a single independent variable and its derivatives. It is considered non-linear if the dependent variable and its derivatives are raised to powers other than 1, or if they are multiplied or divided by each other.

## 2. How do you solve a first-order non-linear ODE?

To solve a first-order non-linear ODE, you can use various methods such as separation of variables, integrating factors, substitution, or numerical methods. The specific method used will depend on the form of the equation and the initial/boundary conditions given.

## 3. What is the importance of solving first-order non-linear ODEs?

First-order non-linear ODEs are used to model many real-world phenomena in fields such as physics, engineering, and biology. By solving these equations, we can gain a better understanding of the behavior and relationships between variables in these systems.

## 4. Can all first-order non-linear ODEs be solved analytically?

No, not all first-order non-linear ODEs can be solved analytically. Some equations may not have closed-form solutions and require numerical methods to approximate a solution. Additionally, some equations may have solutions that cannot be expressed in terms of elementary functions.

## 5. What are some applications of solving first-order non-linear ODEs?

Solving first-order non-linear ODEs has many practical applications, such as predicting the motion of objects under the influence of forces, analyzing chemical reactions, and modeling population growth. It is also used in various engineering fields to design and optimize systems and processes.

• Differential Equations
Replies
3
Views
1K
• Differential Equations
Replies
2
Views
2K
• Differential Equations
Replies
7
Views
2K
• Differential Equations
Replies
4
Views
1K
• Differential Equations
Replies
7
Views
1K
• Differential Equations
Replies
28
Views
2K
• Differential Equations
Replies
5
Views
2K
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
3
Views
2K
• Differential Equations
Replies
2
Views
707