Let y = y(x) be the solution of the differential equation (x - x3)dy = (y + yx2 - 3x4)dx, …

Question

Let y = y(x) be the solution of the differential

equation (x - x3)dy = (y + yx2 - 3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :

Accepted Answer

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

$$ \Rightarrow xdy - ydx = (y{x^2} - 3{x^4})dx + {x^3}dy$$

$$ \Rightarrow {{xdy - ydx} \over {{x^2}}} = (ydx + xdy) - 3{x^2}dx$$

$$ \Rightarrow d\left( {{y \over x}} \right) = d(xy) - d({x^3})$$

Integrate

$$ \Rightarrow {y \over x} = xy - {x^3} + c$$

given f(3) = 3

$$ \Rightarrow {3 \over 3} = 3 \times 3 - {3^3} + c$$

$$ \Rightarrow c = 19$$

$$\therefore$$ $${y \over x} = xy - {x^3} + 19$$

at $$x = 4,{y \over 4} = 4y - 64 + 19$$

$$15y = 4 \times 45$$

$$ \Rightarrow y = 12$$

Solutions · 1