The solution of the differential equation \( \displaystyle\frac{dy}{dx}=\sin x+\cos x \) with \( x=0,\ y=-1 \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( y=\sin x-\cos x \)
The equation can be integrated directly:
\( \displaystyle y=\int(\sin x+\cos x)\,dx \)
\( =\sin x-\cos x+C \)
The condition \( y=-1 \) at \( x=0 \) gives \( -1=0-1+C \), so \( C=0 \).
Hence \( y=\sin x-\cos x \).
Tip: The condition \( y(0)=-1 \) is satisfied by both Options (1) and (4), so the differential equation itself has to be used to choose between them.