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Calculus, SPEC2 2024 VCAA 7 MC

A solution to the differential equation

\(\dfrac{d y}{d x}=e^{x-y}(\cos (x-y)-\cos (x+y))\)  can be found using

  1. \(\displaystyle \int e^y \cos (y) d y=2 \int e^x \cos (x) d x\)
  2. \(\displaystyle\int \frac{e^y}{\sin (y)} d y=2 \int e^{-x} \sin (x) d x\)
  3. \(\displaystyle\int \frac{e^y}{\sin (y)} d y=2 \int e^x \sin (x) d x\)
  4. \(\displaystyle\int e^{-y} \sin (y) d y=2 \int \frac{e^x}{\cos (x)} d x\)
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\(C\)

Show Worked Solution

\(\cos(x-y)-\cos(x+y)\)

\(=[\cos(x)\cos(y)+\sin(x)\sin(y)]-[\cos(x)\cos(y)-\sin(x)\sin(y)]\)

\(=2\sin(x)\sin(y)\)
 

\(\dfrac{d y}{d x}\) \(=e^{x-y}(\cos (x-y)-\cos (x+y))\)  
  \(=e^{x-y} \times 2\sin(x)\sin(y)\)  
  \(=2e^{x}\sin(x) \left(\dfrac{\sin(y)}{e^{y}}\right) \)  

 
\(\displaystyle \int \dfrac{e^{y}}{\sin(y)}\,dy=\displaystyle \int 2e^{x}\sin(x)\,dx\)

\(\Rightarrow C\)