Calculus question, plz help

Question

Using Euler's Method with a step size of h = 0.3, estimate the value for y(0.9) for the differential equation 3 times dy dx plus 5 times y squared equals the sine of x, with y(0.3) = 5.

Answer 1

3y' + 5y^2 = sin x

3y' = sinx - 5 y^2

y' = (1/3) sinx - (5/3)y^2

x y=f(x)

0.3 5

0.36 2.50591040413322679

0.42 1.88499719344441339

0.48 1.53783096057627327

0.54 1.31057413775640859

0.6 1.14909640053387547

0.66 1.0283469962297853

0.72 0.93485957880377177

0.78 0.860651029035084677

0.84 0.800644598041177864

0.9 0.751434283203343123

Answer 2

Euler method uses y(x+h) = y(x) + h y'(x)

We can get y(.9) from y(.3) in two steps of h=.3

y'=(1/3)[sin(x)-5y²]

hy' = .3y' =0.1(sin(x) -5y²]

y(0.3) = 5

hy'(0.3) =-0.015

y(0.6) = y(0.3) + hy'(0.3) = 4.985

hy'(0.6) =-.124

y(0.9) = y(.6) + hy'(.6) = 4.861