Animated Solution for Mathematics - Differential Equations: Let $y=y(x)$ be the solution of the differential equation $(x^2+4{)}^{2}dy+(2x^{3}y+8xy-2)dx=0$. If $y(0)=0$, then $y(2)$ is equal to

Imagine you are standing on the precipice of a complex problem. You see the differential equation $(x^2+4{)}^{2}dy+(2x^{3}y+8xy-2)dx=0$ and it feels like a tangled knot.

But in mathematics, as in life, complexity is often just simplicity waiting to be revealed. Our goal is to find $y(2)$ given $y(0)=0$. Let us take a breath and untangle this together.

The Recognition

First, we must bring order to chaos. We divide the entire equation by $dx$ to get $\frac{dy}{dx}$.

Then, we group the terms involving $y$. By factoring out $y$ from the terms $2x^{3}y$ and $8xy$, we see the structure:

Now, to reach the standard form $\frac{dy}{dx}+P(x)y=Q(x)$, we divide by $(x^2+4{)}^2$. Notice how $2x^3+8x$ is just $2x(x^2+4)$.

This allows us to simplify the coefficient of $y$ to $P(x)=\frac{2x}{x^2+4}$. Our equation is now elegantly poised:

The Integrating Factor

Now we introduce the hero of this story: the Integrating Factor ($IF$). The formula $IF=e^{\int P(x)dx}$ is our bridge.

Substituting $P(x)=\frac{2x}{x^2+4}$, we face the integral $\int \frac{2x}{x^2+4}dx$. Look closely—the numerator is the derivative of the denominator!

This is a classic substitution, yielding $\ln (x^2+4)$. Thus, $IF=e^{\ln (x^2+4)}$, which simplifies beautifully to $x^2+4$. This is the magic multiplier that will allow us to integrate the left side as a product rule derivative.

The Integration

With our $IF$ in hand, the general solution is $y\cdot IF=\int (Q(x)\cdot IF)dx+C$. Substituting our values, we get:

The terms cancel out, leaving us with the integral of $\frac{2}{x^2+4}$. This is a standard form: $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}{\tan }^{-1}(\frac{x}{a})$.

With $a=2$, the integral becomes:

We have $y(x^2+4)={\tan }^{-1}(\frac{x}{2})+C$. Using the initial condition $y(0)=0$, we find $0(4)={\tan }^{-1}(0)+C$, so $C=0$.

Our particular solution is $y(x^2+4)={\tan }^{-1}(\frac{x}{2})$. Finally, to find $y(2)$, we substitute $x=2$:

This simplifies to $8y(2)={\tan }^{-1}(1)=\frac{\pi }{4}$. Dividing by 8, we get the final result:

$$y(2)=\frac{\pi }{32}$$

We have navigated the complexity and arrived at the elegant truth. Well done!