Problem with constants in differential equation?

Question

Assuming the following differential equation:

$$y'+y=4x^3y$$

I proceeded in this way

$$\frac{dy}{dx} = y(4x^3-1)$$ $$\int\frac{dy}{y} = \int(4x^3-1)dx$$ $$lny = x^4 - x + c$$

My solution $$y = e^{x^4-x+c}$$ But book solution is $$y = ke^{x^4-x}$$

What's wrong exactly?

Answer 1

Nothing is wrong.

Note that the exponential you have in your solution, can be written as :

$$e^{x^4-x+c} = e^{x^4-x}e^c$$

by applying the exponential rule : $e^{a+b}=e^ae^b$

Now, if you let $k=e^c$, which is a constant $k\in \mathbb R$ since $e^c$ is constant too as $c$ is a constant, you get the solution given by your book :

$$y(x) = ke^{x^4-x}$$

Answer 2

Nothing is wrong, look at the following: $$y = e^{x^4-x+c}= e^{x^4-x}e^c= e^ce^{x^4-x}$$ now set $k=e^c$, you can do it because both $e$ and $c$ are just constants, therefore $e^c$ is also a constant.

With this you get $$y = e^{x^4-x+c}=ke^{x^4-x}$$