Animated Solution for Mathematics - Differential Equations: Let $y=y(x)$ be the solution of the differential equation $e^x\sqrt{1-y^2}dx+\left(\frac{y}{x}\right)dy=0,y(1)=-1$. Then the value of $(y(3){)}^2$ is equal to:

Welcome, future engineers and mathematicians! Today, we are going to embark on a journey through a beautiful differential equation.

Often, when students see an equation like $e^x\sqrt{1-y^2}dx+(\frac{y}{x})dy=0$, they feel a sense of dread. They see mixed variables, square roots, and exponential functions, but this is not a monster; it is a puzzle waiting for you to reveal its hidden order.

The first step is the separation of variables. We start with the given equation:

Our goal is to isolate $dy$ with $y$ and $dx$ with $x$. By rearranging, we get:

Now, we divide both sides by $\sqrt{1-y^2}$ and multiply by $x$ to obtain the elegant form:

The variables are now perfectly separated, with the left side being a function of $y$ and the right side a function of $x$.

Now that we have separated the variables, we must integrate both sides:

For the left side, we use substitution. Let $u=1-y^2$, then $du=-2ydy$, which implies $ydy=-\frac{1}{2}du$. Substituting this into the integral, we get:

For the right side, we use integration by parts with $u=x$ and $dv=e^{x}dx$. Applying the formula $\int udv=uv-\int vdu$, we get:

Combining our results, we obtain the general solution:

We are given the initial condition that the curve passes through the point $(1,-1)$. Substituting $x=1$ and $y=-1$ into our equation:

The constant vanishes, leaving us with the particular solution:

We are at the finish line. We need to find $(y(3){)}^2$. Substituting $x=3$ into our particular solution:

To isolate $(y(3){)}^2$, we square both sides:

Rearranging the terms, we find the final result:

You have mastered the separation, the integration, and the application of initial conditions. Keep this confidence with you as you tackle the next problem!