Exercise · ZW-ORDER-AND-LINEARITY-008 · ★★☆ Intermediate
For which values of \(n\) is \(y'+y=y^n\) linear?
Choose one option.
Show answer
- B. only \(n=0\) and \(n=1\)
Explanation
With \(n=0\) it becomes \(y'+y=1\) and with \(n=1\) it becomes \(y'=0\); both are linear. For other \(n\), \(y^n\) is not linear.
Review the concept: Order and linearity
The order of a differential equation is that of its highest derivative; the equation is linear if the unknown and its derivatives appear only to the first power and are never multiplied together.
\[a_n(x)\,y^{(n)}+\cdots+a_1(x)\,y'+a_0(x)\,y=g(x)\]
More practice problems: order and linearity
- ZW-ORDER-AND-LINEARITY-001 What is the order of y'''+2y'=x?
- ZW-ORDER-AND-LINEARITY-002 What is the order of (y')^3+y=0?
- ZW-ORDER-AND-LINEARITY-003 Which of the following equations is linear?
- ZW-ORDER-AND-LINEARITY-004 Which of the following equations is NOT linear?
- ZW-ORDER-AND-LINEARITY-005 What is the order of (y'')^2+y'=x?
- ZW-ORDER-AND-LINEARITY-006 Classify x^2y''+e^xy'+y=0:
- ZW-ORDER-AND-LINEARITY-007 Classify y''+y y'=0:
- ZW-ORDER-AND-LINEARITY-009 In a linear equation, the coefficients of y,y',dots may depend on: