( X + X) Dy Dx = Y + Y

3 min read Jun 16, 2024

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

This article will guide you through the process of solving the differential equation (x + x) dy/dx = y + y. We'll break down the steps and explain the underlying concepts.

Simplifying the Equation

First, let's simplify the given equation:

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

This simplifies to:

2x dy/dx = 2y

We can further simplify by dividing both sides by 2:

x dy/dx = y

Separating Variables

Now, let's separate the variables to make the equation easier to solve. To do this, we'll move all the y terms to the left side and all the x terms to the right side:

dy/y = dx/x

Integrating Both Sides

Next, we integrate both sides of the equation. Remember, the integral of 1/x is ln|x| and the integral of 1/y is ln|y|:

∫ dy/y = ∫ dx/x

This gives us:

ln|y| = ln|x| + C

Where C is the constant of integration.

Solving for y

To find the explicit solution for y, we need to get rid of the natural logarithm. We can do this by exponentiating both sides of the equation:

e^(ln|y|) = e^(ln|x| + C)

This simplifies to:

|y| = e^(ln|x|) * e^C

Since e^C is a constant, we can replace it with another constant, let's call it K:

|y| = K|x|

Finally, we can remove the absolute value by considering both positive and negative cases:

y = Kx or y = -Kx

Conclusion

We have successfully solved the differential equation (x + x) dy/dx = y + y and obtained the general solution: