Math Problem Statement
find a function y(x) such that 2yy'=x and y(2)=3
Solution
To solve the differential equation with the initial condition , follow these steps:
Step 1: Separate the variables
The given differential equation is:
First, rewrite as :
Now, separate the variables to get all terms involving on one side and terms involving on the other:
Step 2: Integrate both sides
Integrate both sides:
The integrals are straightforward:
Step 3: Solve for
Take the square root of both sides:
Step 4: Apply the initial condition
We are given that . Substituting and into the equation:
Square both sides to eliminate the square root:
Solve for :
Step 5: Final solution
Substitute into the equation for :
Since , we choose the positive square root. Therefore, the solution is:
Final answer:
Would you like more details or have any questions?
Here are 5 related questions:
- How do you solve separable differential equations in general?
- Can you explain the method of verifying a solution to a differential equation?
- How do initial conditions affect the choice of the constant in solutions?
- What is the difference between explicit and implicit solutions to differential equations?
- How does the solution change if the initial condition were different?
Tip: Always check if a solution satisfies both the differential equation and the initial conditions when solving.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Separation of Variables
Initial Conditions
Formulas
2yy' = x
y = √((x²/2) + C)
Theorems
Fundamental Theorem of Calculus
Separable Differential Equations
Suitable Grade Level
Undergraduate Math (Calculus I/II)
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