Answer:
f(x) = 2e^(x^3/3) +1
Step-by-step explanation:
This is called a "separable" differential equation, because the variables can be separated:
[tex]\dfrac{dy}{y-1}=x^2\cdot dx[/tex]
Now, both sides can be integrated.
[tex]\displaystyle \int{\dfrac{dy}{y-1}}=\int{x^2}\,dx\\\\\ln(y-1)=\dfrac{x^3}{3}+c_1\\\\y-1=c_2\cdot e^{x^3/3}\qquad c_2\text{ is a free constant different from $c_1$}\\\\3=c_2e^0+1\qquad\text{apply initial condition to find $c_2=2$}\\\\\textsf{Add 1 to get y alone. Use the found value of $c_2$. Replace $y$ with $f(x)$.}\\\\\boxed{f(x)=2e^{x^3/3}+1}[/tex]
