AP EAMCET · Maths · Determinants
If \(f(x), f^{\prime}(x) f^{\prime \prime}(x)\), are positive functions and \(f(0)=1, f^{\prime}(0)=2\), then the solution of the differential equation \(\left|\begin{array}{ll}f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x)\end{array}\right|=0\) is
- A \(e^{2 x}\)
- B \(2 \sin x+1\)
- C \(\sin ^2 x+2 x+1\)
- D \(e^{4 x}\)
Answer & Solution
Correct Answer
(A) \(e^{2 x}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {Since, }\left|\begin{array}{ll} f(x) & f^{\prime}(x) \\ f^{\prime}(x) & f^{\prime \prime}(x) \end{array}\right|=0 \\ & \Rightarrow \quad f(x) f^{\prime \prime}(x)-\left(f^{\prime}(x)\right)^2=0 \\ & \Rightarrow \quad f(x) f^{\prime…
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