Question

Find the solution of the differential equation that satisfies the given initial condition.

Answer 1

`y = √(x^2+4)`

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

`\int\limits {x} \, dx = \int\limits{y} \, dy\\(x^2)/(2) = (y^2)/(2) \\ (y^2)/(2) = (x^2)/(2) + C\\y^2 = x^2 + 2C\\y^2 = x^2 + K; K = 2C`

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:

`y^2+x^2 = K\\(-2)^2+0^2 = K\\4 = K`

Step 4: Substitute K = 4 into the resulting differential equation above

`y^2=x^2+4\\y = √(x^2+4)`

This gives the solution to the differential equation.