Math Problem Statement

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Solution

The equation given in the image is:

sinx1+ydydx=cosx\frac{\sin x}{1 + y} \cdot \frac{dy}{dx} = \cos x

This is a first-order differential equation. Let's go step by step to solve it.

Step 1: Separate the variables

We want to separate yy and xx on different sides. The equation can be rewritten as:

sinx1+ydy=cosxdx\frac{\sin x}{1 + y} \cdot dy = \cos x \cdot dx

Now, separate the variables:

dy1+y=cosxsinxdx\frac{dy}{1 + y} = \frac{\cos x}{\sin x} \cdot dx

Step 2: Simplify the right-hand side

Notice that:

cosxsinx=cotx\frac{\cos x}{\sin x} = \cot x

So the equation becomes:

dy1+y=cotxdx\frac{dy}{1 + y} = \cot x \cdot dx

Step 3: Integrate both sides

Now, integrate both sides:

11+ydy=cotxdx\int \frac{1}{1 + y} \, dy = \int \cot x \, dx

The integrals are straightforward:

  • The left-hand side integrates to ln1+y\ln |1 + y|.
  • The right-hand side integrates to lnsinx\ln |\sin x|.

Thus, we have:

ln1+y=lnsinx+C\ln |1 + y| = \ln |\sin x| + C

Step 4: Solve for yy

Exponentiate both sides to get rid of the logarithms:

1+y=Asinx1 + y = A \sin x

where A=eCA = e^C is a constant. Finally, solve for yy:

y=Asinx1y = A \sin x - 1

Final solution

The general solution to the differential equation is:

y=Asinx1y = A \sin x - 1

where AA is an arbitrary constant.

Would you like more details or have any further questions? Here are some related questions:

  1. What is a first-order differential equation?
  2. How do you handle separable differential equations?
  3. Can you explain the technique of integrating logarithmic and trigonometric functions?
  4. Why do we use the constant A=eCA = e^C after integrating?
  5. How can we find the particular solution if initial conditions are given?

Tip: Always look for opportunities to separate variables in a differential equation to simplify the solution process.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Separation of Variables
Trigonometric Functions
Integration

Formulas

Separation of variables: dy/dx = f(x)g(y)
Trigonometric identity: cot(x) = cos(x) / sin(x)
Integration of logarithmic and trigonometric functions

Theorems

Separation of Variables Method
Fundamental Theorem of Calculus

Suitable Grade Level

College/University Level (Calculus I or II)