Question

Solve dy/dx = sqrt x+16 subject to the initial condition y(0)=0

Answer 1

`y=(2)/(3) x^{(3)/(2)} + 16x`

or

`y=(2)/(3) \sqrt{x^(3)} + 16x`

Explanation:

`(dy)/(dx) = √(x) + 16\\ \int\limits {dy} \ =\int\limits { (√(x) + 16)} \, dx =\\=\int\limits { (x^{(1)/(2) } + 16)} \, dx=\int\limits {x^{(1)/(2) } } \, dx +\int\limits{16} \, dx = (2)/(3) x^{(3)/(2)} + 16x + C\\y=(2)/(3) x^{(3)/(2)} + 16x + C\\0=(2)/(3) *0^{(3)/(2)} + 16*0 + C\\C=0\\y=(2)/(3) x^{(3)/(2)} + 16x`