Solve the differential equation.

Question

_x² _dy/dx=2y

answer choice

A) y=-2/x +c

B) y=Ce^2/x

C) y=Ce^-2/x

D) y=Ce^-2x

Answer 1

Hi Adrian!

So this given differential equation can be converted into a separable ordinary differential equation by separating the x and y terms to either side

1. x² dy /dx = 2y

Taking x and dx to right side and shifting 2y to the left hand side we get :

1. dy / 2y = dx / x²

Integrating both sides of the equation, we get

1. (1/2) ln (y) = (-1 / x) + C (integral of 1 / y = ln (y) and 1/ x² is -1 / x using the power rule of integrals)

Now we take (1/2) to the right hand side and to remove the ln (natural logarithm) , we take exponential (e) on both sides and we get:

1. y = e^{(-2/x + 2c)}
2. y = e^(-2/x) * e^2C

Now considering e^2C as a new constant, we get the final answer as

1. y = C₁e^(-2/x)

Hence the correct option is (C)

Hope this helps!