Question

In Problemsexpress the solution of the given initial-value problem in terms of an integraldefined function.

$\mathbf{x}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{\left(}\mathbf{sinx}\mathbf{\right)}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10}$

Short answer

The solution is $y\left(x\right)=10e^{-\int \frac{\sin x}{x}\text{d}x}$.

Step-by-step solution

Definition

A first-order differential equation of the form $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathit{y}\mathbf{\right)}$is said to be separable or to have separable variables.

separate variables

The given differential equation, can be written as

$\begin{array}{rcl}\frac{\text{d}y}{\text{d}x}& =& -\frac{y}{x}\sin x\\ \frac{\text{d}y}{y}& =& -\frac{\sin x}{x}\text{dx Separate variables.}\end{array}$

Integrating

Integrating both sides we get:

$\begin{array}{rcl}\int \frac{\text{d}y}{y}& =& -\int \frac{\sin x}{x}\text{d}x\text{Integrate each side.}\\ \ln \left|y\right|& =& -\int \frac{\sin x}{x}\text{d}x+c\\ y& =& e^{-\int \frac{\sin x}{x}\text{d}x}{e}^c \left(e^{a+b}=e^a{e}^b\right)\\ y& =& e^{-\int \frac{\sin x}{x}\text{d}x}{c}_1 \left(e^c=c_1\right)\end{array}$

Apply initial condition

Use the initial condition $y\left(0\right)=10$, to get $c_1$.

$c_1=10$

Substitute the value $c_1=10$.

$y\left(x\right)=10e^{-\int \frac{\sin x}{x}\text{d}x}$