Question

Find the general solution of the following equation. Express the solution explicitly as a function of the independent variable.

Answer 1

Answer:

Therefore the general solution is

$2 \sqrt w = 2 ln(x) - 5 \frac1x +c$

Step-by-step explanation:

Integration Rule:

1. $\int x^n dx= \frac{x^{n+1}}{n+1}+c$
2. $\int \frac1x dx= ln(x) +c$

Given differential equation is

$x^2 \frac{dw}{dx}= \sqrt{w}(2x+5)$

$\Rightarrow x^2 dw= \sqrt{w} (2x+5) dx$    [ multiplying dx both sides]

$\Rightarrow \frac{dw}{\sqrt w}= \frac{(2x+5)}{x^2} dx$                [ dividing $x^2\sqrt w$ both sides]

Integrating both sides

$\int \frac{dw}{\sqrt w}=\int \frac{(2x+5)}{x^2} dx$

$\Rightarrow \int w^{-\frac12} dw=\int (\frac{2x}{x^2}+\frac{5}{x^2} )dx$

$\Rightarrow \int w^{-\frac12} dw=\int \frac{2}{x}dx +\int\frac{5}{x^2} dx$

$\Rightarrow \frac{w^{-\frac12+1}}{-\frac12+1} =2ln x+5 \frac{x^{-2+1}}{-2+1}+c$   [ c is arbitrary constant]

$\Rightarrow 2 \sqrt w = 2 ln(x) - 5 \frac1x +c$

Therefore the general solution is

$2 \sqrt w = 2 ln(x) - 5 \frac1x +c$