Exercise · ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-004 · ★☆☆ Basic
Solve \(y'=\dfrac{y}{x}\).
Choose one option.
Show answer
- D. \(y=Cx\)
Explanation
With \(v=y/x\): \(v+xv'=v\), so \(v'=0\), \(v=C\) and \(y=Cx\).
Review the concept: First-order homogeneous equation
A first-order homogeneous equation is a differential equation in which \(y'\) is a function \(F\) of the quotient \(y/x\).
\[\frac{dy}{dx}=F\!\left(\frac{y}{x}\right),\quad v=\frac{y}{x}\;\Rightarrow\; x\,v'=F(v)-v\]
More practice problems: first-order homogeneous equation
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-001 Which of these equations is first-order homogeneous?
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-002 With y=vx, where v depends on x, what is y'?
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-003 Applying y=vx to y'=(x+y)/(x), what does xv' reduce to?
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-005 Find the general solution of y'=(x+y)/(x) for x>0.
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-006 Solve y'=(y^2+xy)/(x^2) for x>0.
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-007 If y'=(x+y)/(x) and y(1)=2, what is y(e)?
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-008 For y'=(x^2+y^2)/(xy), what is F(v) with v=y/x?
- ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-009 Why does the substitution v=y/x work in y'=F(y/x)?
← ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-003 ZW-FIRST-ORDER-HOMOGENEOUS-EQUATION-005 →