7.6 Finding General Solutions Using Separation of Variables

Learning objectives

—FUN-7.D—Determine general solutions to differential equations• FUN-7.D Determine general solutions to differential equations.• FUN-7.D.1 Some differential equations can be solved by separation of variables.• FUN-7.D.2 Antidifferentiation can be used to find general solutions to differential equations.• Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

A first-order equation is separable when it can be written as $dy/dx=f(x)g(y)$ . Rearrange it so one side contains only $y$ and $dy$ and the other only $x$ and $dx$ , then antidifferentiate both sides.

\frac{1}{g(y)},dy=f(x),dx\qquad\Longrightarrow\qquad\int\frac{1}{g(y)},dy=\int f(x),dx+C

Factor the derivative rule into an $x$ part and a $y$ part.
Separate the variables algebraically.
Integrate both sides and use one arbitrary constant.
Solve explicitly for $y$ when useful.
Differentiate the result to verify it.

For $dy/dx=xy$ , first assume $y\ne0$ and write $dy/y=x\,dx$ . Then $\ln|y|=x^2/2+C$ , so $y=Ce^{x^2/2}$ after absorbing sign and magnitude into $C$ . Differentiation gives $y'=Cxe^{x^2/2}=xy$ , confirming the family.

Dividing by $g(y)$ can discard constant solutions where $g(y)=0$ . Here division by $y$ temporarily excluded $y=0$ ; checking the original equation shows it is also a solution, and it is included by allowing $C=0$ . Do not add unrelated constants to both sides—two antiderivative constants combine into one arbitrary constant.

Next Topic

—7.7 Finding Particular Solutions Using Initial Conditions and Separation of VariablesRead