Two first-order differential equations are given below

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Two first-order differential equations are given below. (I) dydx=(x2+1) y\dfrac{dy}{dx}=\bigl(x^2+1\bigr)\,y (II) dydx=x+y\dfrac{dy}{dx}=x+y Answer the following questions about these equations. (1) Describe, in clear mathematical language, the structural condition that allows a first-order differential equation to be rewritten so that all terms involving yy (including dydy) appear on one side of the equation and all terms involving xx (including dxdx) appear on the other. (2) Using the condition you stated in part (a), determine whether equation (I) satisfies that condition. Provide explicit algebraic reasoning in your explanation. (3) Using the same condition, determine whether equation (II) satisfies that condition. Provide explicit algebraic reasoning in your explanation.

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