Question

Help please i don’t know how to do it

Answer 1

f(x) = 2e^(x^3/3) +1

Explanation:

This is called a "separable" differential equation, because the variables can be separated:

`(dy)/(y-1)=x^2\cdot dx`

Now, both sides can be integrated.

`\displaystyle \int{(dy)/(y-1)}=\int{x^2}\,dx\\\\\ln(y-1)=(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}`