Question

If the solution of \(\dfrac{dy}{dx}=\dfrac{y^3\cos\sqrt{x}}{\sqrt{x}e^{1/y^2}}\), \(y(0)=1\), is \(\dfrac{1}{y^2}=\log_e(f(x))\), then \(f(x)=\)

• A. \(4+4\sin\sqrt{x}\)
• B. \(e\sin\sqrt{x}\)
• C. \(1-4\sin\sqrt{x}\)
• D. \(e-4\sin\sqrt{x}\)

Hint:

When a differential equation contains \(e^{1/y^2}\), try the substitution \[ u=\frac{1}{y^2}. \] This often converts the equation into a simpler separable form.

Answer 1

The Correct Option is D

Step 1: Separate variables directly.
dy/dx = y³ cos√x / (√x e^(1/y²)). Rearranging: e^(1/y²) / y³ dy = cos√x / √x dx.

Step 2: Substitute u = 1/y² on the left.
du = –2/y³ dy → e^(1/y²)/y³ dy = –(1/2) e^u du. Right side: let t = √x, dt = dx/(2√x) → cos√x/√x dx = 2 cos t dt.

Step 3: Integrate both sides.
–(1/2)∫ e^u du = 2∫ cos t dt → –(1/2)e^u = 2 sin t + C → e^u = –4 sin√x + C'. Using y(0)=1 → u=1, e = C'. So e^(1/y²) = e – 4 sin√x.

Step 4: Final Answer:
f(x) = e – 4 sin√x.