The solution to a 1st order nonlinear ODE is as follows:
$y=e^{\tan^{-1}{x}}(A+\int^xe^{-(tan^{-1}a)/3}\frac{a}{3}da)^3$
Applying boundary condition: $y(0) = 1$
We get $y=e^{\tan^{-1}{x}}(1+\int_0^xe^{-(tan^{-1}a)/3}\frac{a}{3}da)^3$
Could someone please explain the intermediate steps. The integral is impossible to evaluate and I dont understand how we can just magically change the integration bounds and change the coefficient to 1 and call it quits....
let's put in an explicit lower bound $c$ and try to satisfy the boundary condition.
$$y(0)=(A+\int^0_ce^{-(tan^{-1}a)/3}\frac{a}{3}da)^3=1$$
$$\implies A+\int^0_ce^{-(tan^{-1}a)/3}\frac{a}{3}da =1$$
we have 2 arbitrary parameters $A$ and $c$ to satisfy this one condition. The parameters are not independent , they must satisfy $$\implies A=1-\int^0_ce^{-(tan^{-1}a)/3}\frac{a}{3}da$$
which has an infinite number of solutions, but one obvious one is to make the integral vanish by setting $c=0$ so that $A=1$