Question

Use the method for solving Bernoulli equations to solve the following differential equation. $$ frac{d y}{d x]+frac{y}{x}=9 x⁴ y² $$ Ignoring lost solutions, if any, the general solution is $y=$ (Type an expression using $x$ as the variable.)

Answer 1

To solve the Bernoulli differential equation
`$rac{dy}{dx} + rac{y}{x} = 9x^4y^2$`t into a linear differential equation, which is then integrated to find the solution for $y$ in terms of $x$.

Step-by-step explanation:

The student is asking to solve a differential equation using Bernoulli's method. The differential equation in question is
`frac{d y}{d x} + frac{y}{x} = 9 x^4 y^2`ulli's equation in fluid dynamics. The methodology to solve this involves a substitution to convert it into a linear differential equation.

1. First, we identify the equation as a Bernoulli's equation, which takes the form y' + P(x)y = Q(x)y^n where n is any real number.
2. In our equation, n = 2, and we make the substitution
   `z = y^(1 - n) or z = y^(-1)`}y'.
3. After substitution and rearranging, we integrate to find the solution for z, then we back-substitute to find y.
4. Finally, we arrive at the general solution for y as an expression in terms of x.