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# Solve the following differential equation. xdydx+2y=x2logx - Mathematics and Statistics

Sum

Solve the following differential equation.

x dy/dx + 2y = x^2 log x

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#### Solution

x dy/dx + 2y = x^2 log x

Dividing throughout by x, we get

dy/dx + 2/x y = x log x

The given equation is of the form

dy/dx + py = Q

where, P =2/x and Q = x log x

∴ I.F. =e ^int pdx = e ^2int1/xdx = e^2log|x| = e log |x^2| = x^2

∴ Solution of the given equation is

y (I.F.) = int Q (I.F.) dx + c

∴ yx^2 = int (x log x )x^2 dx + c

= int x^3 log x dx +c

= log x int x^3 dx - int (d/dx log x int x^3 dx) dx+c

= x^4/4 log x - int 1/x (x^4/4) dx +c

= x^4/ 4 logx - 1/4 int x^3 dx +c

∴  yx^2 = x^4/4 logx - X^4/16 +c

Concept: Differential Equations
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Differential Equation and Applications
Exercise 8.5 | Q 1.3 | Page 168
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