Second order differential equation question help?

The differential equation

y'' −ay' +by=0

has a solution y=xe^(5x).

then a =? and b = ? to what??

thanks

1 Answer

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  • 8 years ago
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    y=xe^5x

    y`=x`*e^5x+x*(e^5x)`

    y`=1*e^5x+x*[e^5x*(5x)`]

    y`=e^5x+x*e^5x*5

    y`=e^5x(1+5x)

    y``=(y`)`

    y``=(e^5x)`(1+5x)+e^5x(1+5x)`

    y``=e^5x(5x)`(1+5x)+e^5x*5

    y``=e^5x*5(1+5x)+5e^5x

    y``=5e^5x(5x+2)

    y``-ay`+by=0

    5e^5x(5x+2)-a[e^5x(1+5x)]+bxe^5x=0

    e^5x[5(5x+2)-a(1+5x)+bx]=0

    25x+10-a-5ax+bx=0

    25x-5ax+bx+10-a=0

    x(25-5a+b)+10-a=0

    25-5a+b=0 (1)

    and

    10-a=0 => a=10

    a=10 in (1) => 25-5*10+b=0 => 25-50+b=0 => -25+b=0 => b=25

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