charleechristan

charleechristan

27.05.2021 •

Mathematics

Please I need help on this one some can help me?

Please I need help on this one some can help me?

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Ответ:

Felicitybuckman

02.12.2020

$f(x)=Ce^{(1+3x^5)^{\frac{4}{3}}}$

The explicit solution of the initial value problem=y= $1.1e^{(1+3x^5)^{\frac{4}{3}}}$

Step-by-step explanation:

We are given that first order separable equation

$\frac{dy}{dx}=20x^4y(1+3x^5)^{\frac{1}{3}}$

$\int \frac{dy}{y}=20\int x^4(1+3x^5)^{\frac{1}{3}}dx$

Suppose 1+3x^5=t

Differentiate w.r.t x

15x^4dx=dt

$x^4dx=\frac{1}{15}dt$

Substitute the values

$ln y=\frac{20}{15}\int t^{\frac{1}{3}}dt$

$ln y=\frac{4}{3}\times \frac{3}{4} t^{\frac{4}{3}}$ +C

By using the formula $\int x^n dx=\frac{x^{n+1}}{n+1}$ +C

$\int \frac{dx}{x}=ln x+C$

$ln y=(1+3x^5)^{\frac{4}{3}}$ +C

$y=e^{(1+3x^5)^{\frac{4}{3}}+C}$

By using identity : $ln x=y\implies x=e^y$

$y=e^{(1+3x^5)^{\frac{4}{3}}}\cdot e^C=Ce^{(1+3x^5)^{\frac{4}{3}}}$

By using identity : $x^a\cdot x^y=x^{a+y}$

Where e^C=C

$y=Ce^{(1+3x^5)^{\frac{4}{3}}}$

By compare with y=Cf(x)

We get $f(x)=Ce^{(1+3x^5)^{\frac{4}{3}}}$

We are given that y(0)=3

Substitute the value

3=Ce

$C=\frac{3}{e}=1.1$

Substitute the value

$y=1.1e^{(1+3x^5)^{\frac{4}{3}}}$

The explicit solution of the initial value problem=y= $1.1e^{(1+3x^5)^{\frac{4}{3}}}$

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