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Let y=y(x) be the solution of the differential equation ( x)1/2 dy= ( 3x-( x)3/2y )dx, 0<x< 2, y ( 4 )= 6 25. If y ( 3 )= 45 , then 4 equals .
48
Dividing by ( x)1/2dx: dydx+( x) y= 3x x (linear)
I.F. =e x dx= x
y x= 4x x dx
Put x=t, 2x dx=dt: 1+t2 t dt=2t1/2+ 25t5/2+C
y x=2( x)1/2+ 25( x)5/2+C
x= 4: 6 25 2=2+ 25+C C=0
x= 3 ( x= 3, x=2): 2y=2 31/4+ 25 35/4= 165 31/4
y= 85 31/4= 45 =2 31/4
4=16 3=48
Answer: 48
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