Answer: [tex]\frac{u^{2}}{2} + \frac{u^{5}}{5} = - \frac{5}{t} + \frac{t^{3}}{3} + C[/tex]
Step-by-step explanation:
We have the following differential equation:
[tex]\frac{du}{dt} = \frac{5 + t^{4}}{ut^{2} + u^{4} t^{2}}[/tex]
Let's begin by applying common factor [tex]t^{2}[/tex] at the numerator and denominator:
[tex]\frac{du}{dt} = \frac{t^{2}(\frac{5}{t^{2}} + t^{2})}{t^{2}(u + u^{4})}[/tex]
Simplifying and separating variables:
[tex](u + u^{4}) du = (\frac{5}{t^{2}} + t^{2}) dt[/tex]
Integrating both sides of the equation and keeping in mind [tex]\int{x^{n}} dx = \frac{x^{n+1}}{n+1} + C[/tex]:
[tex]\int{u+u^{2}} du = \int{\frac{5}{t^{2}} + t^{2}} dt[/tex]
We have the following result:
[tex]\frac{u^{2}}{2} + \frac{u^{5}}{5} = - \frac{5}{t} + \frac{t^{3}}{3} + C[/tex]
[tex]\dfrac{u^2}{2}+\dfrac{u^5}{5} = (\dfrac{-5}{t}+\dfrac{t^3}{3}) +C[/tex]
Step-by-step explanation:
Given :
[tex]\dfrac{du}{dt} = \dfrac{(5+t^4)}{(ut^2+u^4t^2)}[/tex]
Calculation :
[tex]\dfrac{du}{dt} = \dfrac{(5+t^4)}{t^2(u+u^4)}[/tex]
[tex]\int (u+u^4)du =\int \dfrac{5+t^4}{t^2}dt[/tex]
[tex]\int (u+u^4)du = \int (\dfrac{5}{t^2}+t^2)dt[/tex]
Now, integrating both the we get
[tex]\dfrac{u^2}{2}+\dfrac{u^5}{5} = (\dfrac{-5}{t}+\dfrac{t^3}{3}) +C[/tex]
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https://brainly.com/question/16240546?referrer=searchResults
