Exercise · ZW-SEPARABLE-EQUATION-002 · ★☆☆ Basic
When separating variables in \(y'=2xy\), which equality of integrals results?
Choose one option.
Show answer
- C. \(\int\frac{dy}{y}=\int 2x\,dx\)
Explanation
Divide by \(y\) and multiply by \(dx\) so that only \(y\) is on the left and only \(x\) on the right. Integrating \(2xy\) with respect to \(x\) without separating is a typical mistake, since \(y\) is not constant.
Review the concept: Separable equation
A separable equation is a first-order differential equation whose derivative is the product of a function \(g\) of \(x\) alone and a function \(h\) of \(y\) alone.
\[\frac{dy}{dx}=g(x)\,h(y)\;\Longrightarrow\;\int\frac{dy}{h(y)}=\int g(x)\,dx\]
More practice problems: separable equation
- ZW-SEPARABLE-EQUATION-001 Which of these equations is separable?
- ZW-SEPARABLE-EQUATION-003 Find the general solution of y'=3y.
- ZW-SEPARABLE-EQUATION-004 Separating variables in y'=x/y, which implicit relation do the solutions satisfy?
- ZW-SEPARABLE-EQUATION-005 Solve the problem y'=y, y(0)=2.
- ZW-SEPARABLE-EQUATION-006 Solve y'=-2xy, y(0)=3.
- ZW-SEPARABLE-EQUATION-007 Solve y'=y^2, y(0)=1.
- ZW-SEPARABLE-EQUATION-008 When solving y'=y(y-2) we divide by y(y-2). Which constant solutions might be lost?
- ZW-SEPARABLE-EQUATION-009 Solve y'=-x/y, y(0)=2.