Question

A Bernoulli differential equation is one of the form

d y/d x+P(x) y=Q(x) yⁿ
Observe that, if n=0 or 1 , the Bernoulli equation is linear. For other values of n, the substitution
u=y¹⁻ⁿransforms the Bernoulli equation into the linear equation
du/dx+(1−n)P(x)u=(1−n)Q(x). Use an appropriate substitution to solve the equation y'-(8/x)y=y³/x¹¹
and find the solution that satisfies y(1)=1

Answer 1

Bernoulli differential equations are transformed into linear equations using an appropriate substitution. The given equation is solved using the substitution u = y^-2, leading to the solution of y after integrating and manipulating the result to fit the initial condition.

Step-by-step explanation:

The student's question involves solving a Bernoulli differential equation of the form dy/dx + P(x) y = Q(x) y^n. When n ≠ 0 or 1, the standard approach to solve it is the substitution u = y^{1-n}, transforming it into a linear equation.

For the given equation y' - (8/x)y = y^3/x^11 with conditions y(1)=1, the substitution is u = y^{1-3} = y^{-2}. This gives us u' + (16/x)u = 1/x^11. Solving this linear equation will lead to the solution for u, and thus for y.

After finding the solution for u, we can retrieve the solution for y and check that it satisfies the initial condition y(1)=1 to confirm the solution.