Question

In Exercises \(1-10\) , use separation of variables to solve the initial value problem. Indicate the domain over which the solution is valid. \(\frac{d y}{d x}=-2 x y^{2}\) and \(y=0.25\) when \(x=1\)

Short answer

The solution to the initial value problem is \( y = -\frac{1}{x^{2}-\frac{1}{4}} \), valid over the domain \( x ≠ ±\frac{1}{2} \).

Step-by-step solution

Arrange the equation

Rearrange the given differential equation \(\frac{d y}{d x}=-2 x y^{2}\) such that all terms involving \( y \) are on one side and all terms involving \( x \) are on the other. This yields \( \frac{1}{y^{2}} \frac{dy}{dx}=-2x\).

Integrate both sides

Integrating both sides of the equation yields \( \int \frac{1}{y^{2}} dy= -2 \int x dx \). Applying the integral rules, this simplifies to \( -\frac{1}{y} = -x^{2} + C \), where \( C \) is the constant of integration.

Apply the initial conditions

Substitute the initial conditions \( y=0.25 \) and \( x=1 \) into the equation \( -\frac{1}{y} = -x^{2} + C\). Simplify the equation to find the value of the constant \( C \). Substituting yields \( -\frac{1}{0.25} = -1 + C \), therefore \( C=\frac{1}{4} \). So, the implicit solution of the differential equation is \( -\frac{1}{y} = -x^{2} + \frac{1}{4} \).

Rewrite in terms of y

Rearranging for \( y \), one lands at \( y = -\frac{1}{x^{2}-\frac{1}{4}} \). This function is valid over the domain \( x ≠ ±\frac{1}{2} \).