Differential Equations Solution Solve the separable equation ( dy(x))/( dx) = x^3 y(x), such that (x,y) = (1,2): Divide both sides by y(x): (( dy(x))/( dx))/(y(x)) = x^3 Integrate both sides with respect to x: ? (( dy(x))/( dx))/(y(x)) dx = ? x^3 dx log(y(x)) = x^4/4+c, y(x) = e^(x^4/4+c) y(x) = c e^(x^4/4) Substitute y(1) = 2 into y(x) = c e^(x^4/4): e^(1/4) c = 2 c = 2/e^(1/4) Substitute c = 2/e^(1/4) into y(x) = c e^(x^4/4): y(x) = 2 e^(1/4 (x^4-1)).

1. Differential Equations
Solution
Solve the separable equation ( dy(x))/( dx) = x^3 y(x), such that (x,y) = (1,2): Divide both sides
by y(x): (( dy(x))/( dx))/(y(x)) = x^3 Integrate both sides with respect to x: ? (( dy(x))/(
dx))/(y(x)) dx = ? x^3 dx log(y(x)) = x^4/4+c, y(x) = e^(x^4/4+c) y(x) = c e^(x^4/4)
Substitute y(1) = 2 into y(x) = c e^(x^4/4): e^(1/4) c = 2 c = 2/e^(1/4) Substitute c = 2/e^(1/4)
into y(x) = c e^(x^4/4): y(x) = 2 e^(1/4 (x^4-1))