Solving y' = (y+9x)^2: Methods & Solutions

Thread starter gikiian
Start date Oct 15, 2011

In summary, solving y' = (y+9x)^2 allows us to find the solution to a differential equation, which is used to model real-world situations in various fields. Different methods such as separation of variables and integrating factors can be used to solve this equation, with the appropriate method depending on the given equation and initial conditions. The steps for solving using separation of variables involve isolating and integrating both sides of the equation, while the particular solution can be determined by using initial conditions. However, there are limitations to solving this equation using numerical methods, as they only provide an approximation of the solution and may not work for all equations.

Oct 15, 2011

gikiian

1. The problem statement:

y' = (y+9x)^2The attempt at a solution

I just need to know how do you bring it into the separable form. Tell me numerically or just in words, I won't mind.
Which method do I need to use?

Thanks

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Oct 15, 2011

LCKurtz

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Change the dependent variable y to u with the substitution u = y + 9x.

FAQ: Solving y' = (y+9x)^2: Methods & Solutions

What is the purpose of solving y' = (y+9x)^2?

The purpose of solving y' = (y+9x)^2 is to find the solution to a differential equation. Differential equations are used to model many real-world situations in fields such as physics, engineering, and economics. By finding a solution to the given equation, we can better understand and predict the behavior of the system being modeled.

What are the different methods used to solve y' = (y+9x)^2?

There are several methods that can be used to solve y' = (y+9x)^2, including separation of variables, integrating factors, and substitution. Each method has its own advantages and may be more suitable for certain types of differential equations. It is important to choose the most appropriate method based on the given equation and initial conditions.

Can you explain the steps involved in solving y' = (y+9x)^2 using separation of variables?

Separation of variables involves isolating the variables on opposite sides of the equation and integrating both sides. For y' = (y+9x)^2, we can separate the y and x terms by dividing both sides by (y+9x)^2, resulting in the equation y'/(y+9x)^2 = 1. Then, we can integrate both sides with respect to y, and solve for y to find the general solution.

How do you determine the particular solution for y' = (y+9x)^2 given initial conditions?

To determine the particular solution for y' = (y+9x)^2, we can use the given initial conditions to solve for the arbitrary constant in the general solution. The particular solution will satisfy both the differential equation and the given initial conditions, providing a unique solution for the system being modeled.

Are there any limitations to solving y' = (y+9x)^2 using numerical methods?

Yes, there are limitations to solving y' = (y+9x)^2 using numerical methods. These methods involve approximating the solution, rather than finding an exact solution. Therefore, the accuracy of the solution depends on the step size used in the numerical method. Additionally, some equations may not have a closed-form solution and can only be solved using numerical methods.