Question

In Exercises \(6-11\) solve the initial value problem. $$ y^{\prime}+\left(\frac{1+x}{x}\right) y=0, \quad y(1)=1 $$

Short answer

Solution: The final solution of the initial value problem is \(y(x)=e\cdot e^{C}\cdot e^{-x-\ln|x|}\).

Step-by-step solution

Identify the coefficients of the ODE

In this case, the given ODE is: $$y^{\prime}+\left(\frac{1+x}{x}\right)y=0$$ Here, the coefficient of \(y\) is \(\frac{1+x}{x}\).

Calculate the integrating factor

The integrating factor is given by the formula: $$IF=e^{\int \frac{1+x}{x} dx}$$ Now, let's compute the integral: \begin{align*} \int \frac{1+x}{x} dx &= \int (1+\frac{1}{x}) dx \\ &= \int 1 dx + \int \frac{1}{x} dx \\ &= x + \ln|x| + C \end{align*} So, the integrating factor (IF) is given by: $$IF=e^{x+\ln|x|+C}$$

Multiply the ODE by the integrating factor

Now we multiply the ODE: \(y^{\prime}+\left(\frac{1+x}{x}\right)y=0\) by the integrating factor \(IF=e^{x+\ln|x|+C}\). This results in: $$e^{x+\ln|x|+C}(y^{\prime}+\left(\frac{1+x}{x}\right)y)=0$$

Solve the new ODE and apply the initial condition

We can now rewrite the new ODE as the derivative of a the product of the integrating factor and \(y\): $$\frac{d}{dx}(y\cdot e^{x+\ln|x|+C})=0$$ To find the solution y(x), we can just integrate both sides with respect to x: $$y(x)e^{x+\ln|x|+C}=k$$ Where \(k\) is the constant of integration. Now we can solve for \(y(x)\): $$y(x)=k\cdot e^{-x-\ln|x|-C}$$ Now, apply the initial condition \(y(1)=1\): $$1=k\cdot e^{-1-\ln|1|-C}$$ $$k=e\cdot e^{C}$$

Find the final solution

By substituting the found value of \(k\) back into the expression of \(y(x)\), we obtain the final solution of the initial value problem: $$y(x)=e\cdot e^{C}\cdot e^{-x-\ln|x|}$$

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) play a fundamental role in mathematics, physics, and engineering, describing a wide range of real-world phenomena. An ODE is a mathematical equation that relates some function of one variable to its derivatives. In the context of the initial value problem presented, the function we're interested in is the variable y, which is dependent on the independent variable x.

The problem gives us an ODE of the form:
$$y^{\textprime} + \left(\frac{1+x}{x}\right) y = 0$$
and an initial condition:$$y(1)=1$$
This combination of an ODE and an initial value is what we call an 'initial value problem'. This initial condition anchors our solution to a specific function that passes through the point (1, 1) on the xy-plane. Without this condition, the ODE would have infinitely many solutions corresponding to different curves, since there could be an infinite amount of functions y that satisfy the ODE.

Integrating Factor Method

To solve certain types of linear ODEs, the integrating factor method is a powerful tool. Specifically, it's advantageous when facing a first-order linear ODE that may not be immediately separable. The concept hinges on finding a function, called the integrating factor, which when multiplied by the original ODE, allows the left-hand side to be written as the derivative of a product of functions.

In the given exercise, the integrating factor (IF) is calculated as follows:$$IF=e^{\int \frac{1+x}{x} dx}$$
After computing the integral, we multiply the entire ODE by the integrating factor to make it possible to write the left-hand side of the equation as a derivative of a product, simplifying the equation to:$$\frac{d}{dx}(y \cdot IF) = 0$$
With that, a relatively simple integration yields the function y, multiplied by our integrating factor. The initial condition is then applied to solve for the constant of integration and thereby obtain the specific solution to the initial value problem.

Separable Differential Equations

The initial value problem might also represent a separable differential equation, depending on its form. A separable differential equation is one that can be written as the product of a function of x and a function of y. These equations are solved by separating the variables, moving all y's to one side and all x's to the other side and then integrating both sides.

However, not all ordinary differential equations are separable, including the example we're working with. It's non-separable because the term $$\left(\frac{1+x}{x}\right) y$$[ is not easily divided into a function of x times a function of y. Thus, we cannot apply separability techniques directly to solve it. Instead, we use the integrating factor method, as shown in the exercise, to manipulate the equation into a form that allows for integration and subsequent application of the initial value to find the particular solution.