We are given the differential equation:

\[
4xyy' = x^2 + 4y^2
\]

with the initial condition \( y(1) = 1 \).

### Step 1: Simplify the equation
We start by simplifying the equation. First, divide both sides by 4:

\[
xyy' = \frac{x^2}{4} + y^2
\]

Now, the equation is more manageable.

### Step 2: Solve the differential equation

This is a first-order, non-linear differential equation, which suggests we might try to solve it using separation of variables or some substitution.

Let's try the substitution \( y = vx \), where \( v = \frac{y}{x} \). Differentiating both sides with respect to \( x \), we get:

\[
y' = v + x v'
\]

Now, substitute \( y = vx \) and \( y' = v + x v' \) into the differential equation:

\[
x(vx)(v + x v') = \frac{x^2}{4} + (vx)^2
\]

Simplify the equation:

\[
x^2 v (v + x v') = \frac{x^2}{4} + x^2 v^2
\]

Cancel \( x^2 \) from both sides:

\[
v(v + x v') = \frac{1}{4} + v^2
\]

Simplify further:

\[
v^2 + x v v' = \frac{1}{4} + v^2
\]

Cancel \( v^2 \) from both sides:

\[
x v v' = \frac{1}{4}
\]

### Step 3: Separate variables

We now have a separable equation:

\[
v v' = \frac{1}{4x}
\]

Rewriting it:

\[
v \, dv = \frac{1}{4x} \, dx
\]

Integrate both sides:

\[
\int v \, dv = \int \frac{1}{4x} \, dx
\]

This gives:

\[
\frac{v^2}{2} = \frac{1}{4} \ln|x| + C
\]

Multiply through by 2:

\[
v^2 = \frac{1}{2} \ln|x| + C'
\]

### Step 4: Substitute back \( v = \frac{y}{x} \)

Now, substitute \( v = \frac{y}{x} \):

\[
\left(\frac{y}{x}\right)^2 = \frac{1}{2} \ln|x| + C'
\]

Multiply through by \( x^2 \):

\[
y^2 = x^2 \left( \frac{1}{2} \ln|x| + C' \right)
\]

### Step 5: Apply the initial condition \( y(1) = 1 \)

Substitute \( x = 1 \) and \( y = 1 \) into the equation:

\[
1^2 = 1^2 \left( \frac{1}{2} \ln(1) + C' \right)
\]

Since \( \ln(1) = 0 \), this simplifies to:

\[
1 = C'
\]

So, \( C' = 1 \). Now the solution becomes:

\[
y^2 = x^2 \left( \frac{1}{2} \ln|x| + 1 \right)
\]

### Final Answer:

\[
y = \pm x \sqrt{ \frac{1}{2} \ln|x| + 1 }
\]

Would you like more details on any step or have questions about this process?

Here are some related questions to explore further:
1. How does the substitution method work for solving differential equations?
2. What are some conditions where separation of variables is applicable?
3. Can this problem be solved using a different method, such as an integrating factor?
4. What is the importance of initial conditions in solving differential equations?
5. How does logarithmic differentiation simplify non-linear equations?

**Tip:** In solving differential equations, always check if the equation can be simplified or if substitutions like \( v = \frac{y}{x} \) can reduce complexity.

Learn how to solve the differential equation 4xyy′ = x^2 + 4y^2 with the initial condition y(1) = 1. This detailed step-by-step solution uses the substitution method and separation of variables, suitable for university-level students studying calculus and differential equations.

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