Question

The solution of the differential equation \(\frac{dy}{dx} = 1 + y^2\) is

• A. \(y = \tan x + c\)
• B. \(y = \tan(x+c)\)
• C. \(y = \tan x\)
• D. \(y = -\tan(x+c)\)

Hint:

When solving differential equations, remember that the constant of integration \(c\) is added immediately after integrating. In this case, `tan⁻¹(y) = x + c` becomes `y = tan(x+c)`, which is different from `y = tan(x) + c`. The position of the constant is crucial.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Solve dy/dx + y/x = x², y(1)=1.

Step 2: Key Formula (Alternate):
Integrating factor = e^∫Pdx = e^∫(1/x)dx = x. Multiply and integrate.

Step 3: Detailed Explanation:
d(xy)/dx = x³ → xy = x⁴/4 + c. y(1)=1 → 1=1/4+c → c=3/4. 4xy=x⁴+3.

Step 4: Final Answer:
Solution is 4xy = x⁴+3.