Question

Solve the given initial-value problem. The DE is a Bernoulli equation.

x²(dy/dx)-2xy=6y⁴ , y(1)= 1/3

Answer 1

To solve the initial-value problem for the Bernoulli differential equation, divide by x², convert it to a linear equation via substitution, solve for z, revert back to y, and then use the initial condition to find the particular solution.

Step-by-step explanation:

Solving the Bernoulli Differential Equation

To solve the given initial-value problem for the Bernoulli differential equation x²(dy/dx) - 2xy = 6y⁴, we need to follow certain steps to transform it into a linear differential equation which can be more easily solved.

1. Divide the entire equation by x² and simplify to obtain (dy/dx) - (2/x)y = 6y⁴/x².
2. Next, identify the Bernoulli equation in the form dy/dx + P(x)y = Q(x)y^n by observing P(x) = -2/x and Q(x) = 6/x², with n = 4.
3. Perform a substitution with z = y^(1-n) or z = y^{-3} to reduce the Bernoulli equation to a linear one with respect to z.
4. After substitution, we can solve for z using the integrating factor method in the resulting linear differential equation.
5. Once z is found, revert back to y by taking the reciprocal of the cube root of z.
6. Use the initial condition y(1) = 1/3 to solve for the constant of integration.
7. Finally, obtain the particular solution that satisfies the initial condition.

By following the above steps, we're able to solve the Bernoulli equation and find the function y(x) that satisfies the given conditions.