Find the particular solution of the following differential equation by using the given boundary conditions.
y(1+x)=x(y+1)(dy/dx) when y=1 x=1
To find the particular solution of a differential equation using boundary conditions, we need to follow these steps:
1. Rewrite the differential equation in a standard form.
Given: y(1+x) = x(y+1) * (dy/dx)
First, distribute the x on the right side:
y(1+x) = xy + x(dy/dx)
Next, move all the terms to one side of the equation:
y(1+x) - xy - x(dy/dx) = 0
2. Simplify the equation if possible.
Now let's simplify this equation a bit further:
y(1 - x) - x(dy/dx) = 0
3. Apply the boundary conditions.
The given boundary conditions are y = 1 when x = 1.
To apply the boundary condition, substitute the values into the equation:
1(1 - 1) - 1(dy/dx) = 0
Simplifying this gives:
-1(dy/dx) = 0
dy/dx = 0
4. Integrate to find the particular solution.
Integrate both sides of the equation with respect to x to obtain the particular solution:
∫(dy/dx) dx = ∫0 dx
Integrating, we get:
y = c + 0
Since the derivative of a constant is zero, the particular solution is:
y = c, where c is an arbitrary constant.
Therefore, the particular solution to the given differential equation, satisfying the boundary condition y = 1 when x = 1, is y = 1.