obtain the general solution to the equation.\n\\(\\frac{dy}{dx}=\\frac{y}{x}+7x + 2\\)\nwhat is an…

Explanation:

Step1: Rewrite the equation in standard form

First, rewrite the given first - order linear differential equation $\frac{dy}{dx}=\frac{y}{x}+7x + 2$ as $\frac{dy}{dx}-\frac{1}{x}y=7x + 2$. The general form of a first - order linear differential equation is $\frac{dy}{dx}+P(x)y = Q(x)$, where $P(x)=-\frac{1}{x}$ and $Q(x)=7x + 2$.

Step2: Find the integrating factor

The formula for the integrating factor $\mu(x)$ is $\mu(x)=e^{\int P(x)dx}$. Here, $\int P(x)dx=\int-\frac{1}{x}dx=-\ln|x|=\ln|x^{-1}|$. So, $\mu(x)=e^{\ln|x^{-1}|}=\frac{1}{x}$.

Step3: Multiply the differential equation by the integrating factor

Multiply $\frac{dy}{dx}-\frac{1}{x}y=7x + 2$ by $\frac{1}{x}$ to get $\frac{1}{x}\frac{dy}{dx}-\frac{1}{x^{2}}y = 7+\frac{2}{x}$. The left - hand side is the derivative of $\frac{y}{x}$ by the product rule, i.e., $\frac{d}{dx}(\frac{y}{x})=7+\frac{2}{x}$.

Step4: Integrate both sides

Integrate $\frac{d}{dx}(\frac{y}{x})=7+\frac{2}{x}$ with respect to $x$. $\int\frac{d}{dx}(\frac{y}{x})dx=\int(7+\frac{2}{x})dx$. So, $\frac{y}{x}=7x + 2\ln|x|+C$.

Step5: Solve for y

Multiply both sides by $x$ to obtain the general solution $y(x)=7x^{2}+2x\ln|x|+Cx$.

Answer:

$\mu(x)=\frac{1}{x}$ $y(x)=7x^{2}+2x\ln|x|+Cx$