A) \[{{y}_{3}}+2{{y}_{2}}-{{y}_{1}}=0\]
B) \[{{y}_{3}}+2{{y}_{2}}-35{{y}_{1}}=0\]
C) \[4{{y}_{3}}+5{{y}_{2}}-20{{y}_{1}}=0\]
D) None of these
Correct Answer: B
Solution :
[b] \[y=a+b{{e}^{5x}}+c{{e}^{-7x}}\] ? (i) \[\therefore \] \[{{y}_{1}}=0+5b{{e}^{5x}}-7c{{e}^{-7x}}\] Dividing by \[y{{e}^{5x}}\], we get: \[{{e}^{-5x}}{{y}_{1}}=5b-7c{{e}^{-12x}}\] Again differentiating both sides w.r.t.x, we get \[{{e}^{-5x}}.{{y}_{2}}+{{y}_{1}}(-5){{e}^{-5x}}=0+84c{{e}^{-12x}}\] Dividing by\[{{e}^{-12x}}\]. We get: \[{{e}^{7x}}({{y}_{2}}-5{{y}_{1}})=84c\] Differentiating both sides w.r.t.x, we get \[{{e}^{7x}}({{y}_{3}}-5{{y}_{2}})+({{y}_{2}}-5{{y}_{1}}).7{{e}^{7x}}=0\] \[\Rightarrow {{y}_{3}}+2{{y}_{2}}-35{{y}_{1}}=0\]
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