(2x+y+1)y'=1

Jasonbradley

3 min read Jun 16, 2024

Solving the Differential Equation (2x+y+1)y' = 1

This article will guide you through the process of solving the first-order ordinary differential equation (ODE):

(2x + y + 1)y' = 1

We will use a combination of techniques to find the general solution of this equation.

1. Identifying the Type of ODE

The given equation is a nonlinear first-order ODE because it contains a product of the dependent variable y and its derivative y'. It also does not fall into any standard forms like separable, exact, or linear equations.

2. Transforming the Equation

To solve this equation, we will use a clever substitution to transform it into a more manageable form.

Let:

u = 2x + y + 1

Then:

du/dx = 2 + y'

Solving for y', we get:

y' = du/dx - 2

Now, substitute these values into the original equation:

(2x + y + 1)(du/dx - 2) = 1

Simplifying, we obtain:

u(du/dx - 2) = 1

3. Solving the Transformed Equation

The transformed equation is now a separable ODE:

u(du/dx) = 2u + 1

Separating the variables:

(u/(2u+1))du = dx

Integrating both sides:

∫(u/(2u+1))du = ∫dx

The integral on the left side requires partial fraction decomposition. The result of the integration is:

(1/2)u - (1/4)ln|2u+1| = x + C

where C is the constant of integration.

4. Back-Substituting and Solving for y

Now we need to substitute back u = 2x + y + 1:

(1/2)(2x + y + 1) - (1/4)ln|2(2x + y + 1) + 1| = x + C

Simplifying and solving for y, we get the general solution of the differential equation:

y = -2x - 1 + (1/2)ln|4x + 2y + 3| + 2C

This is the general solution to the given ODE. It represents a family of curves that satisfy the original equation.

5. Conclusion

We have successfully solved the nonlinear first-order ODE (2x + y + 1)y' = 1 using a clever substitution and techniques for solving separable ODEs. The general solution found provides a comprehensive understanding of the behavior of the family of solutions to this particular differential equation.