Uppgift A:8 

restart 

`:=`(ode, {y(0) = 0, diff(y, x) = `+`(`*`(4, `*`(x, `*`(`+`(y, `*`(2, `*`(`^`(x, 2))))))))}) 

{y(0) = 0, diff(y(x), x) = `+`(`*`(4, `*`(x, `*`(`+`(y(x), `*`(2, `*`(`^`(x, 2))))))))} (2.1)
 

`:=`(sol, dsolve(ode)) 

y(x) = `+`(`-`(1), `-`(`*`(2, `*`(`^`(x, 2)))), exp(`+`(`*`(2, `*`(`^`(x, 2)))))) (2.2)
 

taylor(rhs(sol), x = 0, 13) 

series(`+`(`*`(2, `*`(`^`(x, 4))), `*`(`/`(4, 3), `*`(`^`(x, 6))), `*`(`/`(2, 3), `*`(`^`(x, 8))), `*`(`/`(4, 15), `*`(`^`(x, 10))), `*`(`/`(4, 45), `*`(`^`(x, 12))))+O(`^`(x, 13)),x,13) (2.3)
 

assume(n, integer) 

`:=`(y[0], 0) 

0 (2.4)
 

for n from 0 to 4 do `:=`(y[`+`(n, 1)], int(`+`(`*`(4, `*`(x, `*`(`+`(y[n], `*`(2, `*`(`^`(x, 2)))))))), x = 0 .. x)) end do 

 

 

 

 

`+`(`*`(2, `*`(`^`(x, 4))))
`+`(`*`(`/`(4, 3), `*`(`^`(x, 6))), `*`(2, `*`(`^`(x, 4))))
`+`(`*`(`/`(2, 3), `*`(`^`(x, 8))), `*`(`/`(4, 3), `*`(`^`(x, 6))), `*`(2, `*`(`^`(x, 4))))
`+`(`*`(`/`(4, 15), `*`(`^`(x, 10))), `*`(`/`(2, 3), `*`(`^`(x, 8))), `*`(`/`(4, 3), `*`(`^`(x, 6))), `*`(2, `*`(`^`(x, 4))))
`+`(`*`(`/`(4, 45), `*`(`^`(x, 12))), `*`(`/`(4, 15), `*`(`^`(x, 10))), `*`(`/`(2, 3), `*`(`^`(x, 8))), `*`(`/`(4, 3), `*`(`^`(x, 6))), `*`(2, `*`(`^`(x, 4)))) (2.5)
 

plot([rhs(sol), y[5]], x = 1 .. 2, thickness = 2) 

Plot_2d