Question

the general solution of x2y′ 2y = 2e1/x√y is: (a) y1/2 = e1/x ( cx − 1 x ) (b) y1/2 = 1 x e1/x cx (c) y1/2 = e1/x ( c − x x ) (d) y1/2 = xe1/x(cx 1) (e) none of the above

Answer 1

The correct general solution to the given differential equation is option (a), which can be derived by separating the variables, substituting u for the square root of y, integrating, and then substituting back in terms of y.

Step-by-step explanation:

Solution to the Differential Equation

To find the general solution to the differential equation x^2y' + 2y = 2e^{1/x}\sqrt{y}, we will use separation of variables and integrate both sides. Let's solve the equation step by step.

First, divide both sides by x^2\sqrt{y} to separate the variables:

y' + 2y/x^2\sqrt{y} = 2e^{1/x}/x^2

Then, let u = \sqrt{y}, implying y = u^2 and y' = 2u*u'. Substitute these into the equation:

2u*u' + 2u^2/x^2u = 2e^{1/x}/x^2

After simplifying, we get:

u' = e^{1/x}/x^2 - 1/x^2

Integrating both sides with respect to x, we obtain the general solution:

u = e^{1/x}(C - 1/x)

Recall that u = \sqrt{y}, so substituting back in:

\sqrt{y} = e^{1/x}(C - 1/x)

Comparing this solution to the options given, we find that option (a) y^{1/2} = e^{1/x} ( Cx - 1/x ) is the correct general solution after multiplying by x on both sides of the equation within the parentheses.