Answer:
[tex]f(x)=Ce^{(1+3x^5)^{\frac{4}{3}}}[/tex]
The explicit solution of the initial value problem=y=[tex]1.1e^{(1+3x^5)^{\frac{4}{3}}}[/tex]
Step-by-step explanation:
We are given that first order separable equation
[tex]\frac{dy}{dx}=20x^4y(1+3x^5)^{\frac{1}{3}}[/tex]
[tex]\int \frac{dy}{y}=20\int x^4(1+3x^5)^{\frac{1}{3}}dx[/tex]
Suppose [tex]1+3x^5=t[/tex]
Differentiate w.r.t x
[tex]15x^4dx=dt[/tex]
[tex]x^4dx=\frac{1}{15}dt[/tex]
Substitute the values
[tex]ln y=\frac{20}{15}\int t^{\frac{1}{3}}dt[/tex]
[tex]ln y=\frac{4}{3}\times \frac{3}{4} t^{\frac{4}{3}}[/tex]+C
By using the formula [tex]\int x^n dx=\frac{x^{n+1}}{n+1}[/tex]+C
[tex]\int \frac{dx}{x}=ln x+C[/tex]
[tex]ln y=(1+3x^5)^{\frac{4}{3}}[/tex]+C
[tex]y=e^{(1+3x^5)^{\frac{4}{3}}+C}[/tex]
By using identity :[tex]ln x=y\implies x=e^y[/tex]
[tex]y=e^{(1+3x^5)^{\frac{4}{3}}}\cdot e^C=Ce^{(1+3x^5)^{\frac{4}{3}}}[/tex]
By using identity :[tex]x^a\cdot x^y=x^{a+y}[/tex]
Where [tex]e^C=C[/tex]
[tex]y=Ce^{(1+3x^5)^{\frac{4}{3}}}[/tex]
By compare with y=Cf(x)
We get [tex]f(x)=Ce^{(1+3x^5)^{\frac{4}{3}}}[/tex]
We are given that y(0)=3
Substitute the value
[tex]3=Ce[/tex]
[tex]C=\frac{3}{e}=1.1[/tex]
Substitute the value
[tex]y=1.1e^{(1+3x^5)^{\frac{4}{3}}}[/tex]
The explicit solution of the initial value problem=y=[tex]1.1e^{(1+3x^5)^{\frac{4}{3}}}[/tex]
