Question

dy/dx=x^3cosy

Answer 1

$\frac {dy}{dx} = x^3cosy$

By separating variables

$\frac {dy}{cosy}$ = x³dx

=> sec y dy = x³dx

Integrating ,

$\int$ sec y dy = $\int$ x³dx + C

=> ln |sec y + tan y| = $\frac{x^4}{4}$ + C , C is integration constant

This is the general solution

In alternative form the answer can be given as below.

4ln | sec y + tan y | = x⁴ + 4C

=> ln |sec y + tan y |⁴ = x⁴+4C

=> (sec y + tan y)⁴ = e$^{x^4+4C}$

=> (sec y + tan y)⁴ = e^4C. e$^{x^4}$

=> (sec y + tan y)⁴ = A e$^{x^4}$

Where A =e^4C is integration constant

$\mathbf {Answer}$

The general solution of the given differential equation is

(sec y + tan y)⁴ = A e$^{x^4}$ where A is integration constant