Math Problem Statement

Find the explicit general solution to the following differential equation. A prime denotes a derivative with respect to x. y' = 5xy^2

Solution

The given differential equation is:

y=5xy2y' = 5xy^2

This is a separable differential equation, meaning we can solve it by separating the variables and then integrating both sides.

Steps to solve:

  1. Separate variables:
    dyy2=5xdx\frac{dy}{y^2} = 5x \, dx

  2. Integrate both sides:

    • The left side involves 1y2\frac{1}{y^2}, so integrate it as follows: dyy2=5xdx\int \frac{dy}{y^2} = \int 5x \, dx
    • The integral of 1y2\frac{1}{y^2} is 1y-\frac{1}{y}, and the integral of 5x5x is 5x22\frac{5x^2}{2}.

    After integration, we get: 1y=5x22+C-\frac{1}{y} = \frac{5x^2}{2} + C where CC is the constant of integration.

  3. Solve for yy: Multiply both sides by 1-1 to get: 1y=5x22C\frac{1}{y} = -\frac{5x^2}{2} - C

    Finally, take the reciprocal of both sides to solve for yy: y=15x22Cy = \frac{1}{- \frac{5x^2}{2} - C}

Thus, the explicit general solution to the differential equation is:

y=15x22Cy = \frac{1}{- \frac{5x^2}{2} - C}


Would you like more details on any of these steps or have any questions?

5 Related Questions:

  1. How would the solution change if the right-hand side were different, e.g., y=kxy2y' = kxy^2?
  2. What is the significance of the constant CC in the solution?
  3. Can this method be applied to non-separable differential equations?
  4. How would we solve this if initial conditions were given, such as y(0)=1y(0) = 1?
  5. 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)