Question

Find all solutions of the following differential equation. Use lower c for the constant of integration

y′ln(y)+x³y=0
Larger solution
y(x)= ______
Smaller solution y(x)= _____

Answer 1

To find the solutions of the given differential equation, we need to separate the variables and integrate. The solutions of the differential equation are known as logistic curves and cannot be expressed in a simple form using elementary functions. We can only describe the general forms of the solutions in different regions.

Step-by-step explanation:

To find the solutions of the given differential equation, we need to separate the variables and integrate. Rearranging the equation, we have:

y'ln(y) + x³y = 0

Dividing both sides by yln(y), we get:

y'/yln(y) = -x³

Integrating both sides, we have:

∫(1/yln(y)) dy = -∫x³ dx

Taking the integral of the left side requires a special technique called the logarithmic integral, which is beyond the scope of this course. The solutions of the differential equation are known as logistic curves and cannot be expressed in a simple form using elementary functions.

Therefore, we cannot find the exact solutions for y(x). We can only describe the general forms of the solutions in different regions, which are:

In region I and III: y(x) = aexp(-x³/3)

For a larger solution, we choose a positive value for a.

For a smaller solution, we choose a negative value for a.