First-order nonlinear ODE problem help

• nominal
In summary, the conversation is about solving a differential equation y'(t) = (y-5t)/(y+t) with an initial value condition of y(1)=0. The attempt at a solution involved using the equation M(x,y)dx+N(x,y)dy = 0 and determining that it was exact. A solution of g(x,y) = yt + (5/2)t^2 + y^2/2 + C was found, but when plugging in the initial value condition, the resulting equation did not match the given solution. It was suggested to rearrange the equation to separate the y and t terms and then integrate both sides.
nominal

Homework Statement

y'(t) = (y-5t)/(y+t) IVP: y(1)=0

Homework Equations

M(x,y)dx+N(x,y)dy = 0
- or do i use -
y'+p(x)y=q(x)

The Attempt at a Solution

Well I used the first equation (with M and N):
1. first checked that it was exact, by taking the partial of M and N with respect to y and t (respectively), and found that it was exact because both = 1.
2. found a solution g(x,y) = yt + (5/2)t^2 +h(y)
- solved for h(y) (by differentiating and integrating with respect to the other variable)
- and came out with g(x,y) = yt + (5/2)t^2 + y^2/2 + C
3. plugged in my initial value condition, and got C = -5/2
4. Tried plugging in for C, and then solving for y, but that didn't work out
5. Looked at the solution and wasn't even close to the answer: ln[(y 2 + 5t2 )/5] + (2/ 5) arctan[y/( 5t)] = 0

I think I'm doing the whole thing incorrectly.

this is separable, try rearranging so all the y terms are on one side, t terms on the other, then intgerate both sides

1. What is a first-order nonlinear ODE problem?

A first-order nonlinear ODE problem is a type of mathematical problem that involves finding an unknown function that satisfies a first-order nonlinear ordinary differential equation (ODE). This means that the equation contains terms with both the function and its derivatives, making it a nonlinear equation.

2. How do I solve a first-order nonlinear ODE problem?

There are various methods for solving first-order nonlinear ODE problems, including separation of variables, substitution, and integrating factors. Generally, these methods involve manipulating the equation to isolate the function and its derivatives on one side and integrating both sides to find the solution.

3. What is the difference between a linear and a nonlinear ODE problem?

A linear ODE problem is one in which the equation contains only linear terms, meaning that the function and its derivatives are raised to the first power. A nonlinear ODE problem, on the other hand, includes terms with higher powers of the function and its derivatives, making it a more complex problem to solve.

4. Can I use software to solve a first-order nonlinear ODE problem?

Yes, there are many software programs available that can solve first-order nonlinear ODE problems. These programs use numerical methods to approximate the solution and can handle more complex and difficult equations that may be difficult to solve by hand.

5. What are some real-world applications of first-order nonlinear ODE problems?

First-order nonlinear ODE problems have many real-world applications, including in physics, engineering, and biology. They can be used to model the behavior of systems that involve changing quantities over time, such as population growth, chemical reactions, and electrical circuits.

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