A) \[y\,{{\sec }^{3}}x={{\sec }^{2}}x+c\]
B) \[y\,{{\sec }^{2}}x=\sec x+c\]
C) \[y\,\,\sin x=\tan x+c\]
D) None of these
Correct Answer: B
Solution :
\[\frac{dy}{dx}+2y\tan x=\sin x\] is a linear differential equation of the form \[\frac{dy}{dx}+y\,f(x)=g(x)\] \ I.F.\[={{e}^{\int{f(x)dx}}}={{e}^{\int{2\tan x\,dx}}}={{e}^{2\log (\sec x)}}={{e}^{\log {{\sec }^{2}}x}}={{\sec }^{2}}x\] Hence, the solution is \[y\,(\text{I}\text{.F}\text{.)}=\int{g(x)\,\text{I}\text{.F}\text{.}\,dx+c}\] \[y({{\sec }^{2}}x)=\int{\sin x\,{{\sec }^{2}}x\,dx+c}\] Þ \[y{{\sec }^{2}}x=\int{\sec x\,\tan x\,dx+c}\]Þ \[y{{\sec }^{2}}x=\sec x+c\].
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