Question

Find the solution of $$ (2 \sin y-x) \frac{d y}{d x}=\tan y $$ if (a) \(y(0)=0\), and (b) \(y(0)=\pi / 2\).

Short answer

The specific solutions are: (a) for y(0)=0 y(x)=\sin^{-1}(\frac{x+0}{2}) , (b) y(x) =à(\sin^{-1(x) -2}

Step-by-step solution

- Rewrite the Differential Equation

Given the differential equation \[ (2 \sin y - x) \frac{d y}{d x} = \tan y \] Start by rewriting it in a simpler form for separation of variables. Divide both sides by \tan y. \[ (2 \sin y - x) \frac{d y}{d x} = \frac{\sin y}{\cos y} \rightarrow (2 \sin y - x) \cos y \frac{d y}{d x} = \sin y \]},

- Isolate variables

Now, reorganize to isolate \frac{d y}{d x}:\[ (2 \sin y - x) \cos y \frac{d y}{d x} = \sin y \rightarrow \frac{ (2 \sin y - x) \cos y }{\sin y} \frac{d y}{d x} = 1 \rightarrow (2 \cos y \right) - x \cot y \frac{d y}{d x} = 1 \]},

- Separate Variables

Separate the variables for integration: \[ (2 \cos y - x \cot y) dy = dx \]

- Integrate Both Sides

Integrate both sides of the equation:\[ \int (2\cos y - x\cot y) dy = \int dx \]Notice that integrating the left side requires splitting the integral:\[\int 2 \cos y \mathrm{d}y - \int x \cot y \mathrm{d} y \]

- Solve the Integrals

Integrate each part separately:\[ \int 2 \cos y \mathrm{d} y = 2 \sin y \] and \[ \int x \cot y \mathrm{d} y = x \int \cot y \frac{dy}{dx} dy = x \ln|\sin y| \] So the general form becomes:\[ 2 \sin y - x \ln|\sin y| = x + c_1 \]

Key concepts

separation of variables

To solve a differential equation using the separation of variables method, we begin by rearranging the equation to isolate the dependent and independent variables on either side. This technique works well for first-order ordinary differential equations of the form \(\frac{dy}{dx} = g(x)h(y) \). When we separate the variables in the given example, we start with:
\( (2 \sin y - x) \frac{d y}{d x} = \tan y \).
We divide both sides by \tan y \, leading to:
\((2 \sin y - x) \frac{dy}{dx} = \frac{\sin y}{\cos y} \rightarrow (2 \sin y - x) \cos y \frac{dy}{dx} = \sin y \)
Now, our goal is to group all y terms (including \frac{dy}{dx}\textup) on one side and x terms on the other:
\((2 \cos y - x \cot y) dy = dx \).
Once the variables are separately grouped, we can proceed to integrate both sides to find the solution.

integration techniques

After successfully separating the variables, it's time to integrate both sides. This step involves calculating the antiderivatives of both functions. For our example, we have the integral:
\( \int (2 \cos y - x \cot y) dy = \int dx \).
Let's break it into two simpler integrals:
\( \int 2 \cos y dy - \int x \cot y dy \).
Start by integrating \ \int 2 \cos y dy:
\int 2 \cos y dy = 2 \sin y.
Now tackle \int x \cot y dy. For this, recognize that \cot y = \frac{\cos y}{\sin y} \ and use substitution if necessary. It simplifies eventually to
\x \int \frac{cv}{sv}dy.
Combining these steps results in:
\2 \sin y - x \ \ln|\sin y| + c.
The constant of integration \c \is obtained when both sides of the separated equation are integrated.

initial conditions

Each initial condition helps find the constants when constants integrate to provide a more accurate solution fitting a specific scenario.