The general solution of the Bernoulli equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$ is

A) $y ^ { 3 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
B) $y ^ { 2 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
C) $y ^ { - 1 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
D) $y ^ { - 3 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
E) $y ^ { - 2 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$

Question

The general solution of the Bernoulli equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$ is

A) $y ^ { 3 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
B) $y ^ { 2 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
C) $y ^ { - 1 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
D) $y ^ { - 3 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$
E) $y ^ { - 2 } = - \frac { 1 } { 2 } + C e ^ { 3 x ^ { 2 } }$

Quizplus · Accepted Answer

The Bernoulli equation can be solved by substituting $v = y^{-3}$, transforming it into a linear differential equation. Solving this gives the general solution $y^{-3} = -\frac{1}{2} + Ce^{3x^2}$.

The General Solution of the Bernoulli Equation y′+2xy+xy4=0y ^ { \prime } + 2 x y + x y ^ { 4 } = 0y′+2xy+xy4=0

The General Solution of the Bernoulli Equation $y ^ { \prime } + 2 x y + x y ^ { 4 } = 0$