3.8.1—Differential equations

For dy/dx=f(x)g(y), separate into dy/g(y)=f(x)dx and integrate. A linear equation dy/dx+P(x)y=Q(x) uses an integrating factor e^{∫Pdx}.

State where division is legal and retain any constant solutions lost by dividing by g(y). Differentiate the result to verify the form.

dy/dx=xy separates as dy/y=x dx, giving ln|y|=x²/2+C and y=Ae^{x²/2}; y=0 is also included by A=0.

Dividing by y can hide the zero solution, and an integrating factor is not needed for every separable equation.