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प्रश्न
For the differential equation, find the general solution:
`x dy/dx + 2y= x^2 log x`
उत्तर
The given equation
`x dy/dx + 2y = x^2 log x`
or `dy/dx + (2/x)y = x log x`
Comparing with `dy/dx + Py = Q`,
P = `2/x` and Q = x log x
∴ I.F. = `e^(int P dx) = e^(int_x^2 dx)`
`= e^(2 log x) = e^(log x^2) = x^2`
Hence the required solution
∴ y × I.F. = ∫ Q × I.F. dx + C
⇒ y × x2 = ∫ x2 + x log x dx + C
⇒ x2 y = ∫ x3 log x + C
⇒ x2 y = `log x * x^4/4 - int 1/4 * x^4/4 dx + C`
⇒ x2 y = `x^4/4 log x - 1/4 int x^3 dx + C`
⇒ x2 y = `x^4/4 log x - 1/4 xx x^4/4 + C`
⇒ y = `x^2/16 (4 log x - 1) + C/x^2`
Which is the required solution.
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