Question

Find the particular solution to the differential equation \(y^{\prime}=(2 x y)^{2}\) that passes through \(\left(1,-\frac{1}{2}\right)\), given that \(y=-\frac{3}{C+4 x^{3}}\) is a general solution.

Short answer

The particular solution is \( y = -\frac{3}{2 + 4x^{3}} \).

Step-by-step solution

Substitute the point into the general solution

The given point is \((1, -\frac{1}{2})\). Substitute \(x = 1\) and \(y = -\frac{1}{2}\) into the general solution \(y = -\frac{3}{C + 4x^{3}}\). We get: \(-\frac{1}{2} = -\frac{3}{C + 4(1)^3}\).

Simplify the equation

Simplify the equation from Step 1: \(-\frac{1}{2} = -\frac{3}{C + 4}\). This simplifies to \(\frac{1}{2} = \frac{3}{C + 4}\).

Solve for C

Cross-multiply to solve for \(C\): \(1 \cdot (C + 4) = 2 \cdot 3\). This simplifies to \(C + 4 = 6\). Subtract 4 from both sides to get \(C = 2\).

Write the particular solution

Insert the value of \(C\) back into the general solution to find the particular solution. The particular solution is \(y = -\frac{3}{2 + 4x^{3}}\).

Key concepts

Particular Solution

When dealing with differential equations, finding the particular solution is a crucial task. It represents a specific solution derived from the original "general solution" by applying given initial conditions, such as specific points that the solution curve must pass through.

In the problem presented, we are asked to find a particular solution for the differential equation based on the general solution provided: \[y = -\frac{3}{C + 4x^{3}}\]Given the point \((1, -\frac{1}{2})\), we substitute into the general solution to find the constant \(C\). The process ensures that the particular solution precisely fits the point given in the problem. Once \(C\) is determined, we substitute it back into the general solution equation, resulting in the particular solution, \[y = -\frac{3}{2 + 4x^{3}}\]This is a unique curve within the family of solutions described by the general solution.

General Solution

The general solution of a differential equation is a broad description considering all possible solutions without the application of initial or boundary conditions. This type of solution typically includes arbitrary constants, like \(C\), which can take on any value to fit various specific scenarios.

In our differential equation, the general solution given is:\[y = -\frac{3}{C + 4x^{3}}\]This expression forms a family of curves, each member identified by different values of the constant \(C\). These curves provide a comprehensive view of all potential behaviors of the system or process described by the differential equation.
By adjusting \(C\), this solution can adhere to various initial conditions provided in problems, quantitatively narrowing down to a particular solution when specific conditions, like a point the curve passes through, are applied.

Substitution Method

The substitution method is a powerful and straightforward approach used to solve differential equations, especially when initial conditions are provided. This technique involves inserting known values, like a specific point or condition, directly into the solution or equation.

In the context of our problem, we employ the substitution method as follows:

• Insert the value of \(x = 1\) and \(y = -\frac{1}{2}\) into the general solution \(y = -\frac{3}{C + 4x^{3}}\).
• This produces an equation that allows us to solve for \(C\), determining it as \(C = 2\).
• Finally, use this \(C\) in the general solution to formulate the particular solution.

This method simplifies finding the unique solution that matches the given initial condition or point, effectively transforming the general "family of solutions" into a precise answer for the problem at hand.