Exercise · ZW-ORDER-AND-LINEARITY-009 · ★★★ Advanced
In a linear equation, the coefficients of \(y,y',\dots\) may depend on:
Choose one option.
Show answer
- D. only the independent variable \(x\)
Explanation
Coefficients may vary with \(x\); if they depended on \(y\), products of the unknown with itself would appear.
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-008 For which values of n is y'+y=y^n linear?