Answer:
[tex]y = \sqrt{x^2+4}[/tex]
Step-by-step explanation:
The question is incomplete. Here is the complete question.
Find the solution of the differential equation that satisfies the given initial condition. dy/dx = x/y, y(0) = -2
Using the variable separable method.
Step 1: Separate the variables
dy/dx = x/y
x dx = y dy
Step 2: Integrate both sides of the resulting equation
[tex]\int\limits {x} \, dx = \int\limits{y} \, dy\\\frac{x^2}{2} = \frac{y^2}{2} \\ \frac{y^2}{2} = \frac{x^2}{2} + C\\y^2 = x^2 + 2C\\y^2 = x^2 + K; K = 2C[/tex]
Note that the constant of integration added to the side containing x
Step 3: Apply the initial condition y(0) = -2
This means when x = 0, y = -2. From step 2:
[tex]y^2+x^2 = K\\(-2)^2+0^2 = K\\4 = K[/tex]
Step 4: Substitute K = 4 into the resulting differential equation above
[tex]y^2=x^2+4\\y = \sqrt{x^2+4}[/tex]
This gives the solution to the differential equation.
