Use the differential equation and the initial condition to find y.

Question

Use the differential equation  and the initial condition  to find y. ​

Quizplus · Accepted Answer

The given differential equation is:

dy/dx = xy

Separating the variables, we get:

1/y dy = x dx

Integrating both sides, we get:

ln|y| = x^2/2 + C

where C is the constant of integration.

Using the given initial condition, y(0) = 2, we have:

ln|2| = 0 + C

C = ln|2|

Substituting this value of C in the integral equation, we get:

ln|y| = x^2/2 + ln|2|

Using properties of logarithms, we can write:

ln|y| = ln|2| + ln(e^(x^2/2))

ln|y| = ln(2e^(x^2/2))

Taking exponential of both sides, we get:

|y| = 2e^(x^2/2)

Since y(0) > 0, we can drop the absolute value signs and write:

y = 2e^(x^2/2)

Therefore, the solution of the differential equation with the given initial condition is y = 2e^(x^2/2), which corresponds to choice (B).

Use the Differential Equation and the Initial Condition