Math Problem Statement
Solution
The given differential equation is:
This is a separable differential equation, meaning we can solve it by separating the variables and then integrating both sides.
Steps to solve:
-
Separate variables:
-
Integrate both sides:
- The left side involves , so integrate it as follows:
- The integral of is , and the integral of is .
After integration, we get: where is the constant of integration.
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Solve for : Multiply both sides by to get:
Finally, take the reciprocal of both sides to solve for :
Thus, the explicit general solution to the differential equation is:
Would you like more details on any of these steps or have any questions?
5 Related Questions:
- How would the solution change if the right-hand side were different, e.g., ?
- What is the significance of the constant in the solution?
- Can this method be applied to non-separable differential equations?
- How would we solve this if initial conditions were given, such as ?
- What is the general approach to solving differential equations using separation of variables?
Tip:
Always check if a differential equation is separable, as this method simplifies many first-order problems.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Separation of Variables
Integration
Formulas
∫ (1/y^2) dy = ∫ 5x dx
-1/y = (5x^2)/2 + C
Theorems
Separable Differential Equations Theorem
Basic Integration Rules
Suitable Grade Level
College Level (Calculus I/II)
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