(x+ye^(y/x))dx-xe^(y/x)dy=0 Y(1)=0

5 min read Jun 17, 2024

Solving the Differential Equation (x + ye^(y/x))dx - xe^(y/x)dy = 0 with Initial Condition y(1) = 0

This article explores the solution to the given differential equation and its initial condition.

Identifying the Type of Differential Equation

The equation (x + ye^(y/x))dx - xe^(y/x)dy = 0 is a homogeneous differential equation. This is because we can rewrite the equation as:

dy/dx = (x + ye^(y/x)) / (xe^(y/x)) = (1 + (y/x)e^(y/x)) / e^(y/x)

Notice that the right-hand side of the equation is a function of the ratio y/x, indicating it's a homogeneous equation.

Solving the Homogeneous Equation

To solve this, we introduce a new variable:

v = y/x

This implies:

y = vx

Differentiating both sides with respect to x, we get:

dy/dx = v + x(dv/dx)

Now, substitute these expressions for y and dy/dx in the original equation:

v + x(dv/dx) = (1 + ve^v) / e^v

Simplifying the equation:

x(dv/dx) = (1 + ve^v) / e^v - v 
x(dv/dx) = (1 - ve^v) / e^v

Now, this equation is separable:

e^v / (1 - ve^v) dv = dx/x

Integrate both sides:

∫(e^v / (1 - ve^v)) dv = ∫(dx/x)

The integral on the left-hand side requires a substitution:

Let u = 1 - ve^v

Then, du = (-e^v - ve^v)dv = -e^v(1 + v)dv

Therefore, dv = -du / (e^v(1 + v))

Substitute back into the integral:

∫(-du / u(1 + v)) = ∫(dx/x)

Now, the left-hand side can be integrated using partial fractions. After integrating both sides and simplifying, we get:

ln|u| - ln|1 + v| = ln|x| + C

Substituting back for u and v:

ln|1 - ve^v| - ln|1 + (y/x)| = ln|x| + C

Simplifying further:

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

Taking the exponential of both sides:

|(1 - (y/x)e^(y/x)) / (1 + (y/x))| = e^C * |x|

Since e^C is an arbitrary constant, we can replace it with another constant, let's say K:

|(1 - (y/x)e^(y/x)) / (1 + (y/x))| = K|x|

Applying the Initial Condition

Now, we use the initial condition y(1) = 0 to solve for K:

|(1 - (0/1)e^(0/1)) / (1 + (0/1))| = K * |1|

Simplifying:

|1| = K

Therefore, K = 1.

Finding the Explicit Solution

Substituting K = 1 back into the general solution and simplifying, we obtain the explicit solution to the differential equation:

(1 - (y/x)e^(y/x)) / (1 + (y/x)) = x

This is the implicit solution to the given differential equation with the initial condition y(1) = 0. Although finding an explicit solution for y in terms of x might be challenging, this implicit solution provides a complete representation of the curve described by the differential equation.