Exercise · ZW-SUPERPOSITION-PRINCIPLE-004 · ★☆☆ Basic
Why is \(y=0\) always a solution of a linear homogeneous equation?
Choose one option.
Show answer
- C. Because with \(c_1=c_2=0\) the combination is \(0\) and \(L[0]=0\)
Explanation
Taking \(c_1=c_2=0\) in the combination gives \(y=0\), and since \(L\) is linear, \(L[0]=0\). Not every equation (nonhomogeneous ones) admits it.
Review the concept: Superposition principle
It states that if \(y_1\) and \(y_2\) are solutions of a linear homogeneous equation \(L[y]=0\), every linear combination of them with constants \(c_1,c_2\) is also a solution.
\[L[c_1y_1+c_2y_2]=c_1L[y_1]+c_2L[y_2]=0\]
More practice problems: superposition principle
- ZW-SUPERPOSITION-PRINCIPLE-001 If e^t and e^-t solve y''-y=0, which of these functions also solves it?
- ZW-SUPERPOSITION-PRINCIPLE-002 If cos 2t and sin 2t solve y''+4y=0, which of these functions is a solution?
- ZW-SUPERPOSITION-PRINCIPLE-003 If e^2t and e^3t solve y''-5y'+6y=0, what is the value at t=0 of the solution e^2t+4e^3t?
- ZW-SUPERPOSITION-PRINCIPLE-005 If e^t and e^-2t solve y''+y'-2y=0, which value of c makes ce^t+e^-2t equal 3 at t=0?
- ZW-SUPERPOSITION-PRINCIPLE-006 If e^-t and e^-2t solve y''+3y'+2y=0, what is y'(0) for y=2e^-t+e^-2t?
- ZW-SUPERPOSITION-PRINCIPLE-007 y=1 solves y''+y=1. Does y=2 solve that same equation?
- ZW-SUPERPOSITION-PRINCIPLE-008 y=-1/t solves y'=y^2, but y=-2/t does not. Why does superposition fail?
- ZW-SUPERPOSITION-PRINCIPLE-009 y_1 and y_2 solve L[y]=g with g≠0. Which equation does y_1-y_2 solve?
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