Question

The answer above is NOT correct. (1 point) A Bernoulli differential equation is one of the form dx dy ​ +P(x)y=Q(x)y n Observe that, if n=0 or 1 , the Bernoulli equation is linear. For other values of n, the substitution u=y 1−n transforms the Bernoulli eq dx du ​ +(1−n)P(x)u=(1−n)Q(x). Use an appropriate substitution to solve the equation y ′ − x 6 ​ y= x 9 y 3 ​ , and find the solution that satisfies y(1)=1 y(x)=

Answer 1

To solve the Bernoulli differential equation y' - x/6y = x^9y^3, we use the substitution u = y^-2 to transform it into a linear differential equation in u. After solving for u, we revert the substitution to find y in terms of x and apply the initial condition y(1) = 1.

Step-by-step explanation:

The equation y' - x/6y = x9y3 is a Bernoulli differential equation, which is non-linear if the exponent n is neither 0 nor 1. In this case, the Bernoulli equation has n = 3. To solve it, we use the substitution u = y1-n or u = y-2. After the substitution, the differential equation becomes linear with respect to u, and we solve for u before finding y.

To perform the substitution, we differentiate u with respect to x, which gives us du/dx = -2y-3y'. Substituting y = u-1/2 and y' = -1/2 u-3/2du/dx into the original equation, we obtain a linear differential equation in u. After simplifying and integrating, we find u as a function of x. Subsequently, we convert back to find y in terms of x and apply the initial condition y(1) = 1 to find the particular solution.