Question

Trying to check my work:
Solve the DE: dy/dx +4xy = x/y^2

Answer 1

\[y \prime + 4xy = x/y^2\] let z = y^3 => z'/3 = (y^2)*y'\[1/3 z \prime +3xy = x\] \[1/3(z \prime +xz)=x\]

Answer 2

\[z \prime +xz = 3x\] \[let \ u (x)= e ^{∫xdx}\]

Answer 3

\[z \prime +xz = 3x\] \[let \ u (x)= e ^{∫xdx}\]

Answer 4

\[u(x) = e ^{x ^{2}/2}\]

Answer 5

\[z \prime e ^{x ^{2}/2}+xye ^{x ^{2}/2}=3xe ^{x ^{2}/2}\]\[(e ^{x ^{2}/2}z)\prime=3xe ^{x ^{2}/2}\]\[e ^{x ^{2}/2}z = ∫3xe ^{x ^{2}/2}dx\] \[e ^{x ^{2}/2}z = 3e ^{x ^{2}/2} + C\] \[z = 3+C \div e ^{x ^{2}/2}\]

Answer 6

\[y^3 = 3+C \div e ^{x ^{2}/2}\] \[y=\sqrt[3]{C \div e ^{x ^{2}/2}}\]

Answer 7

\[
y' = x/y^2-4xy
\]\[
y' = x(1/y^2-4y)
\]It looks separable, though.

Answer 8

ahh i see it now, \[dy/(1/y^2 - 4y) = xdx\]