3 min read Jun 17, 2024

Solving the Differential Equation: (y^2 + xy^2)y' = 1

This article aims to provide a comprehensive solution to the given differential equation:

(y^2 + xy^2)y' = 1

We will utilize various techniques to arrive at the general solution.

1. Simplifying the Equation

First, let's simplify the equation by factoring out y² on the left-hand side:

y²(1 + x)y' = 1

2. Separating Variables

To solve this differential equation, we need to separate the variables 'x' and 'y'. Divide both sides of the equation by y²(1+x):

y' = 1/(y²(1+x))

Now, we can rewrite y' as dy/dx:

dy/dx = 1/(y²(1+x))

Multiply both sides by dx and y²:

y² dy = dx/(1+x)

3. Integrating Both Sides

Now we have successfully separated the variables. Let's integrate both sides of the equation:

y² dy = ∫ dx/(1+x)

The left side integrates to (y³/3) and the right side integrates to ln|1+x| + C, where C is the constant of integration:

(y³/3) = ln|1+x| + C

4. Solving for y

To isolate y, we can perform the following steps:

  • Multiply both sides by 3: y³ = 3ln|1+x| + 3C
  • Take the cube root of both sides: y = (3ln|1+x| + 3C)^(1/3)

5. Simplifying the Constant

Since 3C is also a constant, we can rewrite the solution as:

y = (3ln|1+x| + C)^(1/3)

where C represents a general constant of integration.


The general solution to the differential equation (y² + xy²)y' = 1 is:

y = (3ln|1+x| + C)^(1/3)

This solution represents a family of curves. The specific curve that satisfies an initial condition can be obtained by plugging in the initial condition and solving for the constant C.

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