Question

Solve the differential equation \(\rm \frac{dy}{dx} + {y\over x} = 4x^2\)

1. x² + c
2. \(\rm x^3 + {c\over x}\)
3. \(\rm x^2 + {c\over x}\)
4. \(\rm x^3 + c\)

Answer 1

Correct Answer - Option 2 : \(\rm x^3 + {c\over x}\)

Concept:

In first order linear differential equation;

\(\rm {dy\over dx}+Py=Q\), where P and Q are function of x

Integrating factor (IF) = e∫ P dx

y × (IF) = ∫ Q(IF) dx

Calculation:

Linear differential equation is of first order

\(\rm \frac{dy}{dx} + {y\over x} = 4x^2\)

Comparing with \(\rm {dy\over dx}+Py=Q\)

So, P = 1/x and Q = 4x²

IF = e∫ \(\rm 1\over x\) dx

IF = e^{ln x}

⇒ IF = x             (∵ e^{ln x} = x)

Now, y × (IF) = ∫ Q (IF) dx

⇒ y × x = ∫ 4x² × x dx

⇒ yx = ∫ 4x³ dx

Integrating,

⇒ yx = x⁴ + c                (where c is integration constant)

⇒ y = \(\boldsymbol{\rm x^3 + {c\over x}}\)