Math Problem Statement
Solution
To solve the initial value problem for the differential equation:
with the initial condition .
Step 1: Integrate to find
Since , we can integrate both sides with respect to to find :
Step 2: Integrate each term separately
-
First term:
- Let , then or .
- Substitute into the integral:
- Substitute back:
-
Second term:
- Let , then or .
- Substitute into the integral:
- Substitute back:
Step 3: Combine results
So, where is the constant of integration.
Step 4: Apply the initial condition
Substitute and :
Calculate each term:
- , so .
- , so .
Thus,
Simplify:
So,
Final Solution
Substitute into the solution:
Let me know if you need further explanation on any steps!
Related Questions
- How would the solution change if the initial condition were different?
- Can this problem be solved using a numerical method? If so, which one?
- How do the solutions behave as in this equation?
- What if the cosine terms were replaced by sine terms in the original differential equation?
- How can we verify this solution by differentiating?
Tip
Remember that integrating terms with trigonometric functions often requires a substitution to simplify the calculation.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differential Equations
Integration
Initial Value Problems
Trigonometric Functions
Formulas
y(x) = ∫(2cos(x/3) + 4cos(3x)) dx
Initial Condition: y(π/2) = 5
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
College Calculus
Related Recommendation
Solve Initial Value Problem for y' = 2 cos(x/3) + 4 cos(3x) with y(π/2) = 5
Solve Initial Value Problem cos(x)y' + sin(x)y = 2cos^3(x)sin(x) - 1
Solve dy/dx = 3x^2 cos(3x^3 + π/2) and find y using integration
Solving dy/dx = (1-y)cosx Differential Equation with Initial Condition y(π) = 2
Solution of y' = 1 + y^2 with Initial Condition y(π/5) = 5 and Evaluation at x = 3π/5