Question

The general solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x y \mathrm{e}^{x^2}\) is

• A. \(y=\mathrm{e}^{-\mathrm{e}^{x^{2}}}\mathrm{c}\), where c is the constant of integration
• B. \(y=\mathrm{e}^{-x^{2}}\mathrm{c}\), where c is the constant of integration
• C. \(y=\mathrm{e}^{\mathrm{e}^{x^{2}}}\mathrm{c}\), where c is the constant of integration
• D. \(y=\mathrm{e}^{x^{2}}\mathrm{c}\), where c is the constant of integration

Hint:

Separate the variables and integrate both sides.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The equation \(\dfrac{dy}{dx}=2xe^{x^2}\,y\) has separable variables.

Step 2: Separate
\(\dfrac{dy}{y}=2xe^{x^2}dx\)

Step 3: Integrate
Put \(t=x^2\), so \(2x\,dx=dt\) and \(\int e^t\,dt=e^{x^2}\). Hence \(\log y=e^{x^2}+\log c\).

Step 4: Solution
\(y=c\,e^{e^{x^2}}\), which is option (C).

Final Answer:
The general solution is y = c e raised to e to the power x squared. \[ \boxed{\text{(C) }y=\mathrm{e}^{\mathrm{e}^{x^{2}}}\mathrm{c}} \]