find the general solution of the following equation. express the solution explicitly as a function of the…

Explanation:

Step1: Separate variables

Separate the variables in the differential - equation $x^{2}\frac{dw}{dx}=\sqrt{w}(3x + 2)$. We get $\frac{dw}{\sqrt{w}}=\frac{3x + 2}{x^{2}}dx$.

Step2: Rewrite the right - hand side

Rewrite $\frac{3x + 2}{x^{2}}$ as $\frac{3}{x}+\frac{2}{x^{2}}$. So the equation becomes $\frac{dw}{\sqrt{w}}=( \frac{3}{x}+\frac{2}{x^{2}})dx$.

Step3: Integrate both sides

Integrate the left - hand side $\int\frac{dw}{\sqrt{w}}=\int w^{-\frac{1}{2}}dw = 2w^{\frac{1}{2}}+C_1$. Integrate the right - hand side $\int(\frac{3}{x}+\frac{2}{x^{2}})dx=3\int\frac{1}{x}dx + 2\int x^{-2}dx=3\ln|x|-\frac{2}{x}+C_2$.

Step4: Solve for $w$

Set $2\sqrt{w}=3\ln|x|-\frac{2}{x}+C$ (where $C = C_2 - C_1$). Then $\sqrt{w}=\frac{1}{2}(3\ln|x|-\frac{2}{x}+C)$. Square both sides to get $w(x)=\left(\frac{3\ln|x|-\frac{2}{x}+C}{2}\right)^{2}$.

Answer:

$w(x)=\left(\frac{3\ln|x|-\frac{2}{x}+C}{2}\right)^{2}$