Respuesta :
Answer:
Therefore the general solution is
[tex]2 \sqrt w = 2 ln(x) - 5 \frac1x +c[/tex]
Step-by-step explanation:
Integration Rule:
- [tex]\int x^n dx= \frac{x^{n+1}}{n+1}+c[/tex]
- [tex]\int \frac1x dx= ln(x) +c[/tex]
Given differential equation is
[tex]x^2 \frac{dw}{dx}= \sqrt{w}(2x+5)[/tex]
[tex]\Rightarrow x^2 dw= \sqrt{w} (2x+5) dx[/tex] [ multiplying dx both sides]
[tex]\Rightarrow \frac{dw}{\sqrt w}= \frac{(2x+5)}{x^2} dx[/tex] [ dividing [tex]x^2\sqrt w[/tex] both sides]
Integrating both sides
[tex]\int \frac{dw}{\sqrt w}=\int \frac{(2x+5)}{x^2} dx[/tex]
[tex]\Rightarrow \int w^{-\frac12} dw=\int (\frac{2x}{x^2}+\frac{5}{x^2} )dx[/tex]
[tex]\Rightarrow \int w^{-\frac12} dw=\int \frac{2}{x}dx +\int\frac{5}{x^2} dx[/tex]
[tex]\Rightarrow \frac{w^{-\frac12+1}}{-\frac12+1} =2ln x+5 \frac{x^{-2+1}}{-2+1}+c[/tex] [ c is arbitrary constant]
[tex]\Rightarrow 2 \sqrt w = 2 ln(x) - 5 \frac1x +c[/tex]
Therefore the general solution is
[tex]2 \sqrt w = 2 ln(x) - 5 \frac1x +c[/tex]
