Math Problem Statement
find the general solution of the differential equation y' = sin x + 2y
Solution
To find the general solution of the differential equation , we will use the method of integrating factors. This is a first-order linear differential equation of the form:
where and .
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Compute the integrating factor :
The integrating factor is given by:
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Multiply the entire differential equation by the integrating factor:
This simplifies to:
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Integrate both sides with respect to :
To integrate , we use integration by parts. Let:
Then,
Now, apply integration by parts:
Substituting and :
Simplify:
To solve , we apply integration by parts again. Let:
Then,
Apply integration by parts again:
Simplify:
Combining the results of both parts, we get:
Solve for :
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Include the constant of integration:
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Solve for :
Therefore, the general solution to the differential equation is:
Would you like any details or have any questions? Here are some related questions you might want to explore:
- How do you solve a differential equation using separation of variables?
- What are homogeneous and non-homogeneous differential equations?
- How do you find particular solutions to non-homogeneous differential equations?
- What are the applications of first-order differential equations in real life?
- How do you solve second-order linear differential equations with constant coefficients?
Tip: When solving differential equations, always ensure that you correctly identify the type of equation and use the appropriate method for the solution.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
First-order linear differential equations
Integrating factors
Integration by parts
Formulas
Integrating factor formula
Integration by parts formula
Theorems
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Suitable Grade Level
College
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