Key Concepts and Formulas

• Homogeneous Differential Equations: A differential equation of the form $\frac{dy}{dx} = f(\frac{y}{x})$ is a homogeneous differential equation.
• Substitution Method: To solve homogeneous differential equations, substitute $y = vx$, where $v$ is a function of $x$. Then $\frac{dy}{dx} = v + x\frac{dv}{dx}$.
• Separation of Variables: A technique used to solve differential equations by isolating variables on opposite sides of the equation and then integrating.

Step-by-Step Solution

Step 1: Rewrite the given differential equation

We are given $y\frac{dy}{dx} = x\left[ \frac{y^2}{x^2} + \frac{\phi(\frac{y^2}{x^2})}{\phi'(\frac{y^2}{x^2})} \right]$ To make it look more like a standard homogeneous differential equation, we divide both sides by $x^2$: $\frac{y}{x} \frac{dy}{dx} = \frac{y^2}{x^2} + \frac{\phi(\frac{y^2}{x^2})}{\phi'(\frac{y^2}{x^2})}$

Step 2: Perform the substitution

Let $v = \frac{y^2}{x^2}$. Then $y^2 = vx^2$. Differentiating both sides with respect to $x$, we get $2y \frac{dy}{dx} = v(2x) + x^2 \frac{dv}{dx}$ Dividing by $2x^2$, we have $\frac{y}{x} \frac{dy}{dx} = v + \frac{x}{2}\frac{dv}{dx}$ Substitute this into the equation from Step 1: $v + \frac{x}{2}\frac{dv}{dx} = v + \frac{\phi(v)}{\phi'(v)}$

Step 3: Simplify and separate variables

Subtracting $v$ from both sides, we have $\frac{x}{2}\frac{dv}{dx} = \frac{\phi(v)}{\phi'(v)}$ Now, separate the variables: $\frac{\phi'(v)}{\phi(v)}dv = \frac{2}{x} dx$

Step 4: Integrate both sides

Integrate both sides with respect to their respective variables: $\int \frac{\phi'(v)}{\phi(v)} dv = \int \frac{2}{x} dx$ The left side integrates to $\ln|\phi(v)|$ and the right side integrates to $2\ln|x| + C$, where C is the constant of integration. Thus, $\ln|\phi(v)| = 2\ln|x| + C$ $\ln|\phi(v)| = \ln|x^2| + C$ Exponentiate both sides: $|\phi(v)| = e^{\ln|x^2| + C} = e^{\ln|x^2|} e^C = x^2 e^C$ Let $K = e^C$, then $\phi(v) = Kx^2$ since $\phi > 0$.

Step 5: Substitute back for v and use the initial condition

Substitute $v = \frac{y^2}{x^2}$ back into the equation: $\phi\left(\frac{y^2}{x^2}\right) = Kx^2$ We are given the initial condition $y(1) = -1$. Substitute $x = 1$ and $y = -1$ into the equation: $\phi\left(\frac{(-1)^2}{1^2}\right) = K(1)^2$ $\phi(1) = K$ So, $K = \phi(1)$. Substituting this back into the equation, we get $\phi\left(\frac{y^2}{x^2}\right) = \phi(1)x^2$

Step 6: Find $\phi \left( {{{{y^2}} \over 4}} \right)$

We want to find $\phi\left(\frac{y^2}{4}\right)$. We need $\frac{y^2}{x^2} = \frac{y^2}{4}$, which means $x^2 = 4$, so $x = 2$ (since $x > 0$). Now substitute $x=2$ into our equation: $\phi\left(\frac{y^2}{4}\right) = \phi(1)(2^2)$ $\phi\left(\frac{y^2}{4}\right) = 4\phi(1)$

Common Mistakes & Tips

• Remember to substitute back for $v$ after integration.
• Pay close attention to the initial conditions and use them to solve for the constant of integration.
• Carefully apply the chain rule when differentiating composite functions during the substitution process.

Summary

We solved the given homogeneous differential equation by using the substitution $v = \frac{y^2}{x^2}$, separating variables, integrating, applying the initial condition to solve for the constant of integration, and then manipulating the equation to find the desired expression. This led us to find that $\phi\left(\frac{y^2}{4}\right) = 4\phi(1)$.

Final Answer

The final answer is \boxed{4\phi(1)}, which corresponds to option (B).