(x+y+1)dy/dx=1 Solution

3 min read Jun 17, 2024

Solving the Differential Equation (x + y + 1) dy/dx = 1

This article will guide you through the process of solving the first-order differential equation (x + y + 1) dy/dx = 1.

Understanding the Equation

This equation is a non-linear, first-order differential equation because:

Non-linear: The terms 'x', 'y', and their product 'xy' are involved in the equation, making it non-linear.
First-order: The highest derivative present is the first derivative, dy/dx.

Solving the Equation

To solve this equation, we'll employ a technique called substitution.

Introduce a new variable: Let u = x + y + 1.
Differentiate u with respect to x: du/dx = 1 + dy/dx.
Rearrange the equation: dy/dx = du/dx - 1.
Substitute into the original equation: (u)(du/dx - 1) = 1.
Simplify: u du/dx - u = 1.
Rearrange: u du/dx = u + 1.
Separate variables: (u/(u+1)) du = dx.
Integrate both sides: ∫(u/(u+1)) du = ∫dx.

To solve the integral on the left side, we can use partial fraction decomposition:

u/(u+1) = 1 - 1/(u+1)

Now, we can integrate:

∫(1 - 1/(u+1)) du = ∫dx
u - ln|u+1| = x + C

Substitute back for u: (x + y + 1) - ln|x + y + 2| = x + C.

Solution

The general solution to the differential equation (x + y + 1) dy/dx = 1 is:

y + 1 - ln|x + y + 2| = C, where C is an arbitrary constant.

Example

Let's find the particular solution that passes through the point (0, 1):

Substituting x = 0 and y = 1 into the general solution, we get:

1 + 1 - ln|0 + 1 + 2| = C

Solving for C:

C = 2 - ln(3)

Therefore, the particular solution passing through (0, 1) is:

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

Conclusion

We successfully solved the differential equation (x + y + 1) dy/dx = 1 using the substitution method and found both the general and particular solutions. This method demonstrates the power of substitution in simplifying complex differential equations.