# What am I doing wrong? Solving first order DE's

• DWill
In summary, the conversation discusses the method of integrating factors to solve a first order differential equation. The individual divides the equation by t and finds the integrating factor to be t*e^t. After taking the integral of both sides, they solve for y and plug in initial values to find the value of C. However, the answer they get does not match the answer given in the book.
DWill
What am I doing wrong?? Solving first order DE's

## Homework Statement

Solve the following first order differential equation:
ty' + (t+1)y = t; y(ln 2) = 1, t > 0

## The Attempt at a Solution

I try to solve this using the method of integrating factors. First I divide the equation by t to get:

y' + [(t+1)/t]y = 1

So following from this I find the integrating factor u(t) = t*e^t

So to solve this I just take an integral of both sides of equations to get:

y*t*e^t = e^t(t-1) + C

Is this right? I now just have to solve for y, and then plug in for the initial values to get C. But the answer I get does not match my book's answer! Can anyone show me what I did wrong here? Thanks

DWill said:
So to solve this I just take an integral of both sides of equations to get:

y*t*e^t = e^t(t-1) + C

Is this right? I now just have to solve for y, and then plug in for the initial values to get C. But the answer I get does not match my book's answer! Can anyone show me what I did wrong here? Thanks

I get the same. What is the book's answer?

The book says the answer is:

y = (t - 1 + 2e^(-t)) / t, where t cannot equal 0.

## 1. What is a first-order differential equation?

A first-order differential equation is a mathematical equation that describes the relationship between an unknown function and its derivative. It typically involves only one independent variable and the derivative of the unknown function with respect to that variable.

## 2. How do I solve a first-order differential equation?

To solve a first-order differential equation, you can use various methods such as separation of variables, integrating factors, or substitution. It is important to remember that not all first-order differential equations have analytical solutions, and numerical methods may be needed to approximate the solution.

## 3. Why do I keep getting different solutions for the same first-order differential equation?

There are many different techniques for solving first-order differential equations, and each method may yield a different solution. Additionally, small variations in the initial conditions or the equation itself can also result in different solutions.

## 4. What are some common mistakes when solving first-order differential equations?

Some common mistakes when solving first-order differential equations include incorrect application of the chosen method, errors in algebraic manipulation, and mistakes in calculating the constant of integration. It is important to double-check your work and be aware of potential pitfalls.

## 5. How can I check if my solution to a first-order differential equation is correct?

You can check the validity of your solution by plugging it back into the original differential equation and verifying that it satisfies the equation. You can also use graphs or tables to compare your solution to known solutions or to confirm that it behaves as expected.

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