If y(x) is the solution of the differential equation x dy - (y² - 4y) dx = 0 for x 0, y(1) = 2, an

Q: Why did we choose the negative sign when removing the absolute value? Could it have been positive?

That is a brilliant question. When we remove the absolute value, we get $\frac{y-4}{y} = \pm Cx^4$. We must check the initial condition $y(1)=2$. Plugging this in, we get $\frac{2-4}{2} = -1$. Since the left side is negative, the right side must also be negative. Thus, we select the negative sign. It is not a guess; it is a mathematical necessity dictated by the initial condition.

Q: Can we use substitution $u = y^2 - 4y$ to solve the integral?

If you try $u = y^2 - 4y$, then $du = (2y - 4) dy$. Notice that we do not have a $(2y - 4)$ term in the numerator to facilitate this substitution. Therefore, standard substitution will not work here. Partial fractions are the most efficient path forward.

Q: What does the condition 'the slope of the curve is never zero' imply?

This condition ensures that the solution curve is well-behaved and does not have stationary points that might complicate the domain or the existence of the inverse function. In this specific problem, it confirms that our solution $y = \frac{4}{1+x^4}$ is valid and continuous for $x > 0$.

Question

If $y(x)$ is the solution of the differential equation $x dy - (y^2 - 4y) dx = 0$ for $x > 0$, $y(1) = 2$, and the slope of the curve $y = y(x)$ is never zero, then the value of $10y(\sqrt{2})$ is ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, JEE aspirants. Today, we are not just solving a problem; we are embarking on a journey through the elegant landscape of differential equations.

When you look at the equation $x dy - (y^2 - 4y) dx = 0$, I want you to see more than just symbols. I want you to see a relationship—a dynamic balance between two variables, $x$ and $y$.

Our mission is to find the specific curve $y(x)$ that satisfies this balance, given the initial condition $y(1) = 2$. Let us begin.

### The Art of Separation

The first step in any differential equation is to ask: Can we separate the variables? Can we isolate the $x$ world from the $y$ world?

Looking at our equation, $x dy = (y^2 - 4y) dx$, the answer is a resounding yes. By dividing both sides by $x$ and by the quadratic expression $(y^2 - 4y)$, we achieve a beautiful separation:

$$ \frac{dy}{y^2 - 4y} = \frac{dx}{x} $$

This is the moment of clarity. We have successfully isolated the variables, turning a complex coupled equation into two independent integration problems.

### The Partial Fraction Magic

Now, we face the integral $\int \frac{dy}{y^2 - 4y}$. The right side, $\int \frac{dx}{x}$, is a standard natural logarithm, but the left side requires a bit of finesse.

We factor the denominator: $y^2 - 4y = y(y-4)$. Now, we use the method of partial fractions. We seek to write $\frac{1}{y(y-4)}$ as $\frac{A}{y} + \frac{B}{y-4}$.

Through simple algebraic manipulation, we find that:

$$ \frac{1}{y(y-4)} = \frac{1}{4} \left( \frac{1}{y-4} - \frac{1}{y} ight) $$

This decomposition is the key that unlocks the door. It transforms a difficult integral into two simple, manageable logarithmic terms.

### Integration and the Constant of Mystery

With our partial fractions in place, the integration becomes a rhythmic process. We have:

$$ \frac{1}{4} \int \left( \frac{1}{y-4} - \frac{1}{y} ight) dy = \int \frac{dx}{x} $$

Integrating both sides, we obtain $\frac{1}{4} (\ln|y-4| - \ln|y|) = \ln|x| + C_1$. Here, $C_1$ is our constant of integration.

It represents the family of all possible curves that satisfy this differential equation. To find our specific curve, we must determine this constant.

Using logarithm properties, we combine the terms: $\ln|\frac{y-4}{y}| = 4\ln|x| + 4C_1$. Exponentiating both sides, we arrive at:

$$ \left| \frac{y-4}{y} ight| = Cx^4 $$

where $C$ is a new constant.

### The Initial Condition

We are given $y(1) = 2$. This is our anchor. We substitute $x=1$ and $y=2$ into our equation:

$$ \left| \frac{2-4}{2} ight| = C(1)^4 $$

This simplifies to $|-1| = C$, so $C=1$. However, we must be careful with the absolute value.

We check the sign: at $x=1, y=2$, the expression $\frac{y-4}{y}$ is $\frac{-2}{2} = -1$. Since the value is negative, we must choose the negative sign when removing the absolute value.

Thus, $\frac{y-4}{y} = -x^4$.

### The Final Reveal

Now, we solve for $y$. Multiplying by $y$, we get $y - 4 = -yx^4$.

Rearranging terms, $y + yx^4 = 4$, which factors to $y(1 + x^4) = 4$. Finally, we have our explicit solution:

$$ y(x) = \frac{4}{1 + x^4} $$

The problem asks for $10y(\sqrt{2})$. We substitute $x = \sqrt{2}$:

$$ y(\sqrt{2}) = \frac{4}{1 + (\sqrt{2})^4} $$

Since $(\sqrt{2})^4 = 4$, the denominator is $1+4=5$. Thus, $y(\sqrt{2}) = \frac{4}{5}$.

Multiplying by $10$, we get $10 	imes \frac{4}{5} = 8$. We have arrived at our destination.

The **final answer is 8**. The beauty of this problem lies not just in the final number, but in the systematic unraveling of the variables, the elegance of partial fractions, and the precision of the initial condition.

Animated Solution for Mathematics - Differential Equations: If $y (x)$ is the solution of the differential equation $x d y - (y^{2} - 4 y) d x = 0$ for $x > 0$ , $y (1) = 2$ , and the slope of the curve $y = y (x)$ is never zero, then the value of $10 y (2)$ is ______.

Enter Numerical Value:

Visualized Solution

Analyze the Differential Equation

Separate the Variables

Setup Integration

Execute Integration

Apply Logarithm Properties

Apply Initial Condition

Calculate Constant C

Rearrange for $y (x)$

Substitute $x = 2$

Final Answer

The Sigma Insight: Variable Separable Method

Analyzing the Setup

The Art of Separation

The Partial Fraction Magic

Integration and the Constant of Mystery

The Initial Condition

The Final Reveal

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