# -b.2.2.1 de with u subst y' + (1/x)y=\sin x; x>0,

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• karush
In summary, the general solution for the given differential equation is $\frac{\sin(x)}{x}-\cos(x)+\frac{c}{x}$, with $c$ being a constant of integration. The book provides the answer as $\frac{c}{x}+\frac{\sin x}{x}-\cos x$, which is equivalent to the general solution. The integrating factor is $\mu(x)=x$.
karush
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$\textsf{Find the general solution of the given differential equation(book answer in red)}$
$$y^\prime + (1/x)y=\sin x \quad x>0, \qquad \color{red} {\frac{c}{x}+\frac{\sin x}{x}-\cos x}$$
ok first $u=1/x$ and $x=1/u$ then
$$u(x) = \exp\int u \, du = e^{\ln(u)}=u +c$$
proceed or ?

$\tiny{Elementary Differential Equations And Boundary Value Problems, \\ By: William E. Boyce and Richard C. Diprima \\ 1969, Second Edition}$

Last edited:
We can see the integrating factor is $\mu(x)=x$ and so the ODE will become:

$$\displaystyle \frac{d}{dx}(xy)=x\sin(x)$$

Upon integrating, we get:

$$\displaystyle xy=\sin(x)-x\cos(x)+c_1$$

Hence:

$$\displaystyle y(x)=\frac{\sin(x)}{x}-\cos(x)+\frac{c_1}{x}$$

MarkFL said:
We can see the integrating factor is $\mu(x)=x$ and so the ODE will become:

$$\displaystyle \frac{d}{dx}(xy)=x\sin(x)$$

Upon integrating, we get:

$$\displaystyle xy=\sin(x)-x\cos(x)+c_1$$

Hence:

$$\displaystyle y(x)=\frac{\sin(x)}{x}-\cos(x)+\frac{c_1}{x}$$

wait why would $u(x)=x$

karush said:
wait why would $u(x)=x$

$$\displaystyle \mu(x)=\exp\left(\int\frac{1}{x}\,dx\right)=e^{\ln(x)}=x$$

## 1. What does the equation "-b.2.2.1 de with u subst y' + (1/x)y=\sin x; x>0" represent?

The equation represents a second-order linear differential equation with a substitution of variables. It is used to model a physical system or phenomenon that involves the rate of change of a quantity with respect to another variable.

## 2. What is the purpose of the substitution of variables in this equation?

The substitution of variables helps to simplify the equation and make it easier to solve. It involves replacing the original variables with new ones that are easier to manipulate, without changing the overall behavior of the equation.

## 3. How is this equation solved?

This equation can be solved using various methods, such as the method of undetermined coefficients, variation of parameters, or the Laplace transform. The specific method used depends on the structure and complexity of the equation.

## 4. What does the "x>0" condition mean in this equation?

The "x>0" condition indicates that the equation is only valid for values of x that are greater than 0. This is a restriction that is often used in differential equations to ensure that the solution is physically meaningful.

## 5. What are some real-world applications of this type of differential equation?

This type of differential equation is commonly used in physics, engineering, and other scientific fields to model various phenomena, such as radioactive decay, population growth, and electrical circuits. It can also be used in economics to model changes in supply and demand.

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