The general solution of the first-order differential equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$ is

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

Question

The general solution of the first-order differential equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$ is

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

Quizplus · Accepted Answer

The given differential equation can be rewritten and solved by separating variables or using an integrating factor. The correct solution involves integrating both sides appropriately, leading to $y = x^2 + C\sqrt{x^2 - 1}$, where $C$ is the constant of integration.

The General Solution of the First-Order Differential Equation (x2−1)y′−xy=x3−2x\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x(x2−1)y′−xy=x3−2x

The General Solution of the First-Order Differential Equation $\left( x ^ { 2 } - 1 \right) y ^ { \prime } - x y = x ^ { 3 } - 2 x$