Question Statement

Solve the differential equation:

$(\sin x+\cos x)dx=\cot y\cos xdy$

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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 algebraically rearrange the equation so that all terms involving $x$ and $dx$ appear on one side, while all terms involving $y$ and $dy$ appear on the other side, allowing us to integrate both sides independently.

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Solution

Begin with the given differential equation:

$(\sin x+\cos x)dx=\cot y\cos xdy$

To separate the variables, divide both sides by $\cos x$ (assuming $\cos x\ne 0$):

$\left(\frac{\sin x+\cos x}{\cos x}\right)dx=\cot ydy$

Now integrate both sides of the equation:

$\int \frac{\sin x+\cos x}{\cos x}dx=\int \cot ydy$

Decompose the fraction on the left side by splitting the numerator:

$\int \left(\frac{\sin x}{\cos x}+\frac{\cos x}{\cos x}\right)dx=\int \cot ydy$

Simplify the terms inside the integral. Note that $\frac{\sin x}{\cos x}=\tan x$ and $\frac{\cos x}{\cos x}=1$:

$\int \tan xdx+\int 1dx=\int \cot ydy$

Evaluate each integral using standard formulas:

• $\int \tan xdx=\ln (\sec x)$
• $\int 1dx=x$
• $\int \cot ydy=\ln (\sin y)$

Applying these results and adding the constant of integration $c$:

$\ln (\sec x)+x=\ln (\sin y)+c$

This is the general solution to the differential equation. If an initial condition were provided (such as $y(x_0)=y_0$), you could substitute these values to solve for the specific constant $c$ and obtain the particular solution.

(Note: The solution can also be expressed in explicit form by exponentiating: $\sin y=\sec x\cdot e^{x-c}$, or equivalently $y=\arcsin \left(\frac{e^x}{K\cos x}\right)$ where $K=e^c$.)

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Key Formulas or Methods Used

• Separation of variables: Rearranging $M(x)dx=N(y)dy$ form
• Algebraic decomposition: $\frac{\sin x+\cos x}{\cos x}=\tan x+1$
• Integral of tangent: $\int \tan xdx=\ln (\sec x)+C$ (or $-\ln |\cos x|+C$)
• Integral of cotangent: $\int \cot ydy=\ln (\sin y)+C$ (or $\ln |\sin y|+C$)
• Constant of integration: Added as $+c$ (or $+C$) when performing indefinite integration

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Summary of Steps

1. Separate variables: Divide both sides by $\cos x$ to isolate $x$-terms on the left and $y$-terms on the right
2. Simplify: Break $\frac{\sin x+\cos x}{\cos x}$ into $\tan x+1$
3. Set up integrals: Write $\int (\tan x+1)dx=\int \cot ydy$
4. Integrate: Apply standard integral formulas to get $\ln (\sec x)+x=\ln (\sin y)+c$
5. General solution: The implicit equation $\ln (\sec x)+x=\ln (\sin y)+c$ represents the family of all solutions; substitute initial values if given to find the specific constant $c$