Math Problem Statement
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Solution
The equation given in the image is:
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 and on different sides. The equation can be rewritten as:
Now, separate the variables:
Step 2: Simplify the right-hand side
Notice that:
So the equation becomes:
Step 3: Integrate both sides
Now, integrate both sides:
The integrals are straightforward:
- The left-hand side integrates to .
- The right-hand side integrates to .
Thus, we have:
Step 4: Solve for
Exponentiate both sides to get rid of the logarithms:
where is a constant. Finally, solve for :
Final solution
The general solution to the differential equation is:
where is an arbitrary constant.
Would you like more details or have any further questions? Here are some related questions:
- What is a first-order differential equation?
- How do you handle separable differential equations?
- Can you explain the technique of integrating logarithmic and trigonometric functions?
- Why do we use the constant after integrating?
- 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)
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