# 2.1.7 Find the solution of the given initial value problem.

• MHB
• karush
Thus, in summary, the solution to the given initial value problem is $y(x)=e^{-x}\int_{0}^{x}\frac{e^t}{1+t^2} \, dt$.
karush
Gold Member
MHB
$\tiny{2.1.{7}}$
$$\displaystyle y^\prime +y =\frac{1}{1+x^2}, \quad y(0)=0$$
$\textit{Find the solution of the given initial value problem.}$ \begin{align*}\displaystyle
u(x) &=e^x\\
(e^x y)'&=\frac{e^x}{1+x^2} \\
e^x y&=\int \frac{e^x}{1+x^2}\, dx\\
&=\color{red}
{\displaystyle e^{-x}\int_{0}^{x}\frac{e^t}{1+t^2} \, dt}
\end{align*}
need help with steps why does the answer have $t$ in it
also if y=0 wouldn't it be 1

Last edited:
The integrand:

$$\displaystyle \frac{e^x}{x^2+1}$$

does not have an anti-derivative in elementary terms, and so the solution is given as a definite integral. The dummy variable $t$ is introduced so that a different variable is used than what is in the upper limit in integration, for the sake of good notation.

## 1. What is an initial value problem?

An initial value problem is a type of mathematical equation that involves finding a solution for a function or system of functions, given an initial condition or starting point. It is commonly used in differential equations and other areas of mathematics and science.

## 2. How is the solution of an initial value problem found?

The solution of an initial value problem is found by using a variety of methods, including analytic techniques, numerical methods, and computer simulations. The specific approach depends on the type of equation and the complexity of the problem.

## 3. What does the term "2.1.7" refer to in this problem?

The term "2.1.7" refers to the section and problem number within a larger textbook or resource. It is used to organize and categorize different problems and make it easier to locate specific information.

## 4. Can an initial value problem have more than one solution?

Yes, an initial value problem can have multiple solutions, especially in cases where the equation is nonlinear or the initial conditions are not well-defined. It is important to carefully consider all possible solutions and choose the most appropriate one for the given problem.

## 5. What is the significance of finding the solution to an initial value problem?

The solution of an initial value problem is significant because it allows us to make predictions and understand the behavior of a system over time. It also helps us to analyze and model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.

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