To solve the initial value problem defined by the equation x² dy/dx = y + xy with the initial condition y(1) = 1, we can start by rewriting the equation.
We can rearrange the equation to isolate y/dx:
dy/dx = (y + xy) / x²
Next, we can separate the variables:
dy / (y + xy) = dx / x²
Now, let’s integrate both sides. We can simplify the left-hand side:
Factor out y from the denominator:
dy / (y(1 + x))
Now we integrate:
∫(1/y) dy = ∫(1/(1+x)) dx
These integrals lead to:
ln|y| = ln|1+x| + C
Exponentiating both sides gives us:
y = K(1+x)
where K = e^C is an arbitrary constant. Now, we apply the initial condition to find K.
Given that y(1) = 1:
1 = K(1+1)
1 = 2K
K = 1/2
Substituting K back into the equation for y:
y = (1/2)(1+x)
Thus, the explicit solution to the initial value problem is:
y = (1 + x) / 2