Problem 28 - Essential Calculus

Question

Solve the differential equation. $$ y^{\prime}=\arctan \frac{x}{2} $$

Step-by-Step Solution

Answer

The solution of the differential equation \(y'= \arctan(\frac{x}{2})\) is \(y(x) = \frac{x}{2} \arctan(\frac{x}{2}) - \frac{1}{4}\ln(1 + (\frac{x}{2})^2) + C\).

Step 1: Identify Type of Differential Equation

Identify the type of differential equation. This equation is of the form \(y' = f(x)\), where \(f(x)\) is a given function of \(x\). In this case, \(f(x) = \arctan(\frac{x}{2})\). It is a first-order ordinary differential equation without explicit dependence on \(y\), which can be solved by direct integration.

Step 2: Integrate Both Sides of the Equation

The differential equation can be represented as \(\frac{dy}{dx} = \arctan(\frac{x}{2})\). We now need to integrate both sides of the equation with respect to \(x\). The left-hand side of the equation, when integrated, gives us \(y\).

Step 3: Integrate the Function

Now we integrate the right hand side, which is the function \(\arctan(\frac{x}{2})\), with respect to \(x\). The integral of \(\arctan(\frac{x}{2})\) with respect to \(x\) involves a standard technique of integration for \(\arctan(x)\), which is \(\int \arctan(x) dx = x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C\), where \(\ln\) denotes the natural logarithm, and \(C\) is the constant of integration.

Step 4: Write the Final Solution

From the integration in Step 3, we apply this result to our specific integral of \(\arctan(\frac{x}{2})\), replacing wherever \(x\) is with \(\frac{x}{2}\) in the standard integral. Thus the solution of the differential equation will be \(y(x) = \frac{x}{2} \arctan(\frac{x}{2}) - \frac{1}{4}\ln(1 + (\frac{x}{2})^2) + C\).