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

find the general solution of the following equation. express the solution explicitly as a function of the independent variable. x^2 dw/dx = √w(3x + 5) w(x)= (use parentheses to clearly denote the argument of each function.)

find the general solution of the following equation. express the solution explicitly as a function of the independent variable. x^2 dw/dx = √w(3x + 5) w(x)= (use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Separate variables

Separate the $w$ - terms and $x$ - terms. We have $\frac{dw}{\sqrt{w}}=\frac{3x + 5}{x^{2}}dx$.

Step2: Rewrite the right - hand side

Rewrite $\frac{3x + 5}{x^{2}}$ as $\frac{3}{x}+\frac{5}{x^{2}}$. So the equation becomes $\frac{dw}{\sqrt{w}}=( \frac{3}{x}+\frac{5}{x^{2}})dx$.

Step3: Integrate both sides

Integrate $\int w^{-\frac{1}{2}}dw=\int(\frac{3}{x}+\frac{5}{x^{2}})dx$. The left - hand side integral: $\int w^{-\frac{1}{2}}dw = 2w^{\frac{1}{2}}+C_1$. The right - hand side integral: $\int\frac{3}{x}dx+\int\frac{5}{x^{2}}dx=3\ln|x|-\frac{5}{x}+C_2$. So, $2\sqrt{w}=3\ln|x|-\frac{5}{x}+C$ (where $C = C_2 - C_1$).

Step4: Solve for $w$

Square both sides to get $w(x)=\left(\frac{3\ln|x|-\frac{5}{x}+C}{2}\right)^{2}$.

Answer:

$w(x)=\left(\frac{3\ln|x|-\frac{5}{x}+C}{2}\right)^{2}$