Question

5. Resolva os seguintes problemas de valor inicial (PVI): (2,0)
a) {(dy / dx - 6 x² - 1 = 0 y (0 ) = 5

b) {((d² y ) / ( dx² ) = 2 - 6 x y (0) = 1 y' (0) = 4

Answer 1

$\frac{dy}{dx} - 6 {x}^{2} - 1 = 0 \\ \frac{dy}{dx} = 6 {x}^{2} + 1 \\ \ \\ ∫ dy= \: ∫(6 {x}^{2} + 1)dx$

y(x) =2x³+x+c

y(0)=2.0³+0+c

c=5

y(x) =2x³+x+5

b)

$\frac{ {d}^{2} y}{d {x}^{2} } = 2 - 6x \\ \frac{d}{dx} ( \frac{dy}{dx}) = 2 - 6x \\ \frac{d}{dx}(dy) = (2 - 6x)dx$

$\frac{d}{dx} (∫dy) = ∫(2 - 6x)dx \\ \ \frac{dy}{dx} = 2x- 3 {x}^{2} + c1 \\ dy = (2x - 3{x}^{2} + c1)dx$

y'(0)=2.0-3.0²+c1

c1=4

$\frac{d}{dx} (∫dy) = ∫(2 - 6x)dx \\ \ \frac{dy}{dx} = 2x- 3 {x}^{2} + c1 \\ dy = (2x - 3{x}^{2} + 4)dx$

$∫dy = ∫(2x - 3 {x}^{2} + 4)dx \\ y = {x}^{2} - {x}^{3} + 4x + c2$

y(0)=0²-0³+4.0+c2

c2=1

y(x) =x²-x³+4x+1