SUMMARY
The initial value problem defined by the differential equation $y'=\dfrac{1+3x^2}{3y^2-6y}$ with the condition $y(0)=1$ is solved analytically by separating variables and integrating both sides, yielding the solution $y^3 - 3y^2 - x^3 - x + 2 = 0$. The constant $c$ is determined to be -2 by applying the initial condition. The solution is valid within the interval $|x|<1$, as the denominator $3y^2-6y$ indicates that $y$ cannot equal 0 or 2, which restricts the values of $x$ to avoid these points.
PREREQUISITES
- Understanding of differential equations, specifically separable equations.
- Knowledge of integration techniques for polynomial functions.
- Familiarity with initial value problems and their solutions.
- Ability to analyze the implications of solution intervals based on function behavior.
NEXT STEPS
- Study the method of separation of variables in differential equations.
- Learn about the implications of initial conditions on the solutions of differential equations.
- Explore the concept of solution intervals and restrictions based on function behavior.
- Investigate the graphical interpretation of differential equations and their solutions.
USEFUL FOR
Mathematics students, educators, and professionals involved in solving differential equations, particularly those focusing on initial value problems and their analytical solutions.