Find the particular solution of the differential equation \(\dfrac{dy}{dx} = y\tan x\), given that \(y = 2\) if \(x = 0\).
Find the particular solution of the differential equation \(\dfrac{dy}{dx} = y\tan x\), given that \(y = 2\) if \(x = 0\).
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Solution and Explanation
The variables can be separated. We integrate both sides to get the general solution and then use the given condition to fix the constant.
Step 2: Separate the variables.
\[ \frac{dy}{y} = \tan x\,dx \]
Step 3: Integrate both sides.
The integral of \(\tan x\) is \(\log|\sec x|\).
\[ \log|y| = \log|\sec x| + \log C \]
\[ y = C\sec x \]
Step 4: Use the condition y = 2 at x = 0.
\[ 2 = C\sec 0 = C(1) \]
So \(C = 2\).
Final Answer:
The particular solution is \(y = 2\sec x\). \[ \boxed{y = 2\sec x} \]
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