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證明 If Y1(t) and Y2(t) are two solutions to a linear, homogeneous differential equation then so is y(t)= c1y1(t)+ c2y2(t)
Set the linear, homogeneour diff. eq. be L(y)=0, where L is a linear operator,
ie. L(a y1+b y2)=a L(y1)+ b L(y2).
Now, we know that L(Y1)=0, L(Y2)=0, so
L(c1 Y1+ c2 Y2)=c1 L(Y1)+ c2 L(Y2)= c1*0 + c2*0 =0
ie. the function c1 Y1+ c2 Y2 satisfies L(y)=0
namely, c1 Y2+ c2 Y2 is a solutions of L(y)=0.
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Set the linear, homogeneour diff. eq. be L(y)=0, where L is a linear operator,
ie. L(a y1+b y2)=a L(y1)+ b L(y2).
Now, we know that L(Y1)=0, L(Y2)=0, so
L(c1 Y1+ c2 Y2)=c1 L(Y1)+ c2 L(Y2)= c1*0 + c2*0 =0
ie. the function c1 Y1+ c2 Y2 satisfies L(y)=0
namely, c1 Y2+ c2 Y2 is a solutions of L(y)=0.