Solve the initial value problem: $$ \frac{d y}{d x}=\frac{-0.5}{\cos (y)}, \quad y(0)=\frac{\pi}{4} $$ $$ y(x)= $$

Question

UpStudy AI · Accepted Answer

$$
\textbf{Step 1: Separate variables.}
$$
Given the differential equation
$$
\frac{dy}{dx} = \frac{-0.5}{\cos(y)},
$$
we can rewrite it as
$$
\cos(y) \, dy = -0.5 \, dx.
$$

$$
\textbf{Step 2: Integrate both sides.}
$$
Integrate the left side with respect to $$ y $$ and the right side with respect to $$ x $$:
$$
\int \cos(y) \, dy = \int -0.5 \, dx.
$$
The integrals become:
$$
\sin(y) = -0.5 x + C,
$$
where $$ C $$ is the constant of integration.

$$
\textbf{Step 3: Apply the initial condition.}
$$
The initial condition is
$$
y(0) = \frac{\pi}{4}.
$$
Substitute $$ x=0 $$ and $$ y=\frac{\pi}{4} $$ into the integrated equation:
$$
\sin\left(\frac{\pi}{4}\right) = -0.5 \cdot 0 + C.
$$
Since
$$
\sin\left(\frac{\pi}{4}\right)= \frac{\sqrt{2}}{2},
$$
we get
$$
C = \frac{\sqrt{2}}{2}.
$$

$$
\textbf{Step 4: Write the solution.}
$$
Substituting $$ C $$ back into the integrated equation, we obtain:
$$
\sin(y) = -0.5 x + \frac{\sqrt{2}}{2}.
$$
To solve for $$ y $$, take the arcsine of both sides:
$$
y = \arcsin\left(-0.5 x + \frac{\sqrt{2}}{2}\right).
$$

$$
\boxed{y(x) = \arcsin\left(-0.5 x + \frac{\sqrt{2}}{2}\right)}
$$
$$ y(x) = \arcsin\left(-0.5x + \frac{\sqrt{2}}{2}\right) $$

Solve the initial value problem:

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Calculus Questions