Class 12 Differential Equations · Mcq Single Correct1 Marksdifferential_equations

Class 12 Maths · Differential Equations · Mcq Single Correct #119

Let \(f\) be a nowhere-zero continuously differentiable function on a connected real interval, satisfying \((f(x))^2-4f(x)f\prime(x)+(f\prime(x))^2=0\). Which family gives all such solutions, with \(\lambda\ne0\)?

(A) \(\lambda e^{(2+\sqrt5)x}\)
(B) \(\lambda e^{(2-\sqrt5)x}\)
(C) \(\lambda e^{(2\pm\sqrt3)x}\)
(D) \(\lambda e^{(3-\sqrt3)x}\)

✅ Step-by-Step Solution

Final Answer: Option (C): \(\lambda e^{(2\pm\sqrt3)x}\)

Divide by \(f^2\), allowed because \(f\ne0\), and set \(r=f\prime/f\). Then \(r^2-4r+1=0\), so \(r\in\{2-\sqrt3,2+\sqrt3\}\). Continuity of \(r\) on a connected interval forces one constant root throughout. Integration gives \(f(x)=\lambda e^{rx}\), with \(\lambda\ne0\) and either root.

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Class 12 Maths · Differential Equations · Mcq Single Correct #119

✅ Step-by-Step Solution

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