Question

The general solution of $\left(x \frac{dy}{dx} - y\right) \sin \frac{y}{x} = x^3 e^x$ is

• A. $e^x(x - 1) + \cos \frac{y}{x} + c = 0$
• B. $x e^x + \cos \frac{y}{x} + c = 0$
• C. $e^x(x + 1) + \cos \frac{y}{x} + c = 0$
• D. $e^x x - \cos \frac{y}{x} + c = 0$

Hint:

Whenever you see the expression $(x dy - y dx)$ or $(x y' - y)$, immediately think of dividing by $x^2$ to create the exact derivative $d(y/x)$, or dividing by $y^2$ to create $-d(x/y)$. It is one of the most powerful shortcuts in differential equations.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a first-order differential equation. We need to find its general solution by using an appropriate substitution.

Step 2: Key Formula or Approach:
The presence of $y/x$ suggests a homogeneous-style substitution or directly forming an exact derivative. The quotient rule for differentiation is $\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{x\frac{dy}{dx} - y}{x^2}$.

Step 3: Detailed Explanation:
The given differential equation is:
$$\left(x \frac{dy}{dx} - y\right) \sin \frac{y}{x} = x^3 e^x$$ Divide both sides of the equation by $x^2$:
$$\left(\frac{x \frac{dy}{dx} - y}{x^2}\right) \sin \frac{y}{x} = x e^x$$ Recognize that the term in the parenthesis is the exact derivative of $\frac{y}{x}$:
$$\frac{d}{dx}\left(\frac{y}{x}\right) \sin \left(\frac{y}{x}\right) = x e^x$$ Now, integrate both sides with respect to $x$:
$$\int \sin\left(\frac{y}{x}\right) \frac{d}{dx}\left(\frac{y}{x}\right) dx = \int x e^x dx$$ Let $v = \frac{y}{x}$, so $dv = \frac{d}{dx}\left(\frac{y}{x}\right) dx$:
$$\int \sin v dv = \int x e^x dx$$ Integrate the left side normally, and the right side using integration by parts ($\int u dv = uv - \int v du$ where $u=x, dv=e^x dx$):
$$-\cos v = x e^x - \int e^x dx$$ $$-\cos v = x e^x - e^x + C_1$$ Substitute $v = \frac{y}{x}$ back into the equation:
$$-\cos \frac{y}{x} = e^x(x - 1) + C_1$$ Rearrange the terms to set the equation to zero:
$$e^x(x - 1) + \cos \frac{y}{x} + C = 0$$ (where $C = C_1$)

Step 4: Final Answer:
The general solution is $e^x(x - 1) + \cos \frac{y}{x} + c = 0$, matching option (A).