Question Statement

Solve the differential equation:

$\frac{dy}{dx}+y\tan 2x=0$

subject to the initial condition $y(0)=2$.

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Background and Explanation

This is a first-order ordinary differential equation that can be solved using the method of separation of variables. The goal is to rearrange the equation so that all terms involving $y$ appear on one side and all terms involving $x$ appear on the other side, allowing us to integrate both sides independently.

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Solution

Step 1: Separate the variables

Starting with the given equation: $\frac{dy}{dx}+y\tan 2x=0$

Rearrange to isolate the derivative: $\frac{dy}{dx}=-y\tan 2x$

Divide both sides by $y$ (assuming $y\ne 0$) and multiply by $dx$ to separate the variables: $\frac{dy}{y}=-\tan 2xdx$

Step 2: Integrate both sides

Integrate the left side with respect to $y$ and the right side with respect to $x$: $\int \frac{dy}{y}=-\int \tan 2xdx$

Rewrite $\tan 2x$ as $\frac{\sin 2x}{\cos 2x}$: $\int \frac{dy}{y}=-\int \frac{\sin 2x}{\cos 2x}dx$

To integrate the right side, we use the substitution method. Notice that the derivative of $\cos 2x$ is $-2\sin 2x$. We manipulate the integrand to match this form: $\int \frac{dy}{y}=\frac{1}{2}\int \frac{-2\sin 2x}{\cos 2x}dx$

Now the numerator is the derivative of the denominator, so the integral becomes: $\ln y=\frac{1}{2}\ln (\cos 2x)+\ln c$

Here, $\ln c$ is the constant of integration (written in logarithmic form for easier combination).

Step 3: Solve for $y$

Apply logarithm properties to combine terms. First, use $k\ln a=\ln (a^k)$: $\ln y=\ln (\cos 2x{)}^{1/2}+\ln c$

Then use $\ln a+\ln b=\ln (ab)$: $\ln y=\ln \left(c(\cos 2x{)}^{1/2}\right)$

Exponentiate both sides to eliminate the logarithm: $y=c\sqrt{\cos 2x}$

Step 4: Apply the initial condition

We are given $y(0)=2$, which means when $x=0$, $y=2$. Substitute these values into equation (1):

$2=c\sqrt{\cos 0}$

Since $\cos 0=1$: $2=c\sqrt{1}$ $2=c\cdot 1$ $2=c$

(Note: The original solution writes $2=d$ here, but this is equivalent to finding $c=2$ for the constant of integration.)

Step 5: Write the final solution

Substitute $c=2$ back into equation (1):

$y=2\sqrt{\cos 2x}$

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Key Formulas or Methods Used

• Separation of variables: $\frac{dy}{y}=f(x)dx$
• Integral of $\frac{1}{y}$: $\int \frac{1}{y}dy=\ln |y|+C$
• Integration of tangent: $\int \tan (ax)dx=-\frac{1}{a}\ln |\cos (ax)|+C$ (derived here using substitution)
• Logarithm properties:
  • $k\ln a=\ln (a^k)$
  • $\ln a+\ln b=\ln (ab)$
• Exponentiation: $e^{\ln y}=y$

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Summary of Steps

1. Rearrange the differential equation to separate variables: $\frac{dy}{y}=-\tan 2xdx$
2. Integrate both sides, using substitution ($u=\cos 2x$) on the right side to get $\ln y=\frac{1}{2}\ln (\cos 2x)+\ln c$
3. Simplify using logarithm properties and exponentiate to obtain the general solution: $y=c\sqrt{\cos 2x}$
4. Substitute the initial condition $y(0)=2$ to find the constant: $c=2$
5. Write the particular solution: $y=2\sqrt{\cos 2x}$