Consider . Which statement best justifies that and cannot be isolated on opposite…

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider

\displaystyle \frac{dy}{dx}=y^{2}+x^{2}

. Which statement best justifies that

x

and

y

cannot be isolated on opposite sides by algebraic manipulation? (a) Because

y^{2}

and

x^{2}

have the same degree, the equation is homogeneous, and homogeneous equations cannot be separated by algebraic means. (b) Because the equation is nonlinear in

y

, isolation of variables is impossible for any nonlinear differential equation. (c) Because the right-hand side is a sum of a

y

-only term and an

x

-only term, it cannot be written as a product

f(x)\,g(y)

, so the variables cannot be separated. (d) After attempting to separate, one would have to integrate

\displaystyle\int\!\frac{1}{y^{2}+x^{2}}\,dy

, which still mixes

x

and

y

inside a single integral; therefore the variables cannot be isolated purely by algebra."

Related theory

Browse the problem library