Question

The sine integral function is defined as $\mathbf{Si}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\int }_{\mathbf{0}}^{\mathbf{x}}\frac{\mathbf{sint}}{\mathbf{t}}\mathbf{dt}$, where the integrand is defined to be 1 at x = 0. See Appendix A. Express the solution of the initial-value problem

${\mathbf{x}}^{\mathbf{3}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{2}{\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{=}\mathbf{10}\mathbf{sinx}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{0}$ in terms of Si(x)

Short answer

So, the solution of initial problem is$y\left(x\right)=10x^{-2}\left[Si\right(x)-Si(1\left)\right]$.

Step-by-step solution

Form of linear equation

The linear differential equation is of the form $\frac{\mathrm{dy}}{\mathrm{dx}}+\mathbf{Py}=\mathbf{Q}$, where P and Q are numeric constants or functions in x.

Evaluation

The given differential equation, can be written as

$\frac{dy}{dx}+\frac{2}{x}y=\frac{10}{x^3}sinx$

Linear differential equation of the first order

localid="1668416535098" $y^{\text{'}}+P\left(x\right)y=f\left(x\right)$

Based on (1) and $\left(2\right)$the given differential equation, we obtain

localid="1668416542753" $P\left(x\right)=\frac{2}{x}\mathrm{and}f\left(x\right)=\frac{10}{x^3}\sin x$

Find integrated factor

The integrating factor is given by

$e^{\int P\left(x\right)dx}$

Use the formula (3), to get integrating factor

role="math" localid="1664272742192" $$\begin{gathered} \begin{array}{l}=e^{2\int \frac{1}{x}\mathrm{d}x}\\ =e^{2\ln \left|x\right|}\\ =e^{\ln x^2}\\ =x^2\end{array} \\ Thus,thesolutionisy{x}^2={\int }_0^x\frac{10t^2}{t^3}\sin t\mathrm{d}t+c \end{gathered}$$

Find the value of constant

$\begin{array}{l}y=x^{-2}10{\int }_0^x\frac{sint}{t}dt+c{x}^{-2}\left(Si\left(x\right)={\int }_0^z\frac{sint}{t}\right)\\ y=x^{-2}10Si\left(x\right)+c{x}^{-2}\end{array}$

Use the initial condition y(1) = 0, to get c.

$\begin{array}{l}0=10Si\left(1\right)+c\\ c=-10Si\left(1\right)\end{array}$

Substitute the value in the equation (4).

$y\left(x\right)=10x^{-2}\left[Si\right(x)-Si(1\left)\right]$

So, the required solution is $y\left(x\right)=10x^{-2}\left[Si\right(x)-Si(1\left)\right]$.