Question

Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).

Hint:

Integrals of the form \( \int e^{ax} \cos(bx) dx \) and \( \int e^{ax} \sin(bx) dx \) are cyclic. When using integration by parts, the original integral will reappear after applying the method twice. Memorizing the standard formula for these integrals can save significant time.

Answer 1

Step 1: Understanding the Concept:
This is a first-order differential equation where the variables are already separated. To solve for y, we need to integrate the right-hand side with respect to x. The integral of \( e^x \cos x \) is a standard type that is solved using integration by parts.
Step 2: Key Formula or Approach:
1. The equation can be written as \( dy = e^x \cos x \, dx \).
2. Integrate both sides: \( \int dy = \int e^x \cos x \, dx \).
3. Use integration by parts formula: \( \int u \, dv = uv - \int v \, du \).
4. A standard integral formula is also available: \( \int e^{ax} \cos(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\cos(bx) + b\sin(bx)) + C \).
Step 3: Detailed Explanation or Calculation:
We need to find the integral \( I = \int e^x \cos x \, dx \).
Let's use the standard formula with \( a=1, b=1 \): \[ I = \frac{e^x}{1^2+1^2}(1\cos(x) + 1\sin(x)) + C \] \[ I = \frac{e^x}{2}(\cos x + \sin x) + C \] The solution to the differential equation is \( y = I \).
Alternatively, using integration by parts:
Let \( u = \cos x \) and \( dv = e^x dx \). Then \( du = -\sin x \, dx \) and \( v = e^x \). \[ I = e^x \cos x - \int e^x (-\sin x) \, dx = e^x \cos x + \int e^x \sin x \, dx \] Apply integration by parts to the new integral: Let \( u = \sin x \) and \( dv = e^x dx \). Then \( du = \cos x \, dx \) and \( v = e^x \). \[ I = e^x \cos x + \left( e^x \sin x - \int e^x \cos x \, dx \right) \] The original integral \( I \) reappears on the right side. \[ I = e^x \cos x + e^x \sin x - I \] \[ 2I = e^x(\cos x + \sin x) \] \[ I = \frac{e^x}{2}(\cos x + \sin x) \] So, the solution is \( y = \frac{e^x}{2}(\cos x + \sin x) + C \).
Step 4: Final Answer:
The solution of the differential equation is \( y = \frac{e^x}{2}(\cos x + \sin x) + C \).