find the general solution of the following equations 1 x2y2x

find the general solution of the following equations:

1. x²y\'\'+2xy\'-12y=0

2. x²y\'\'+xy\'=0

3.x²y\'\'+xy\'-y=0

Solution

1. x2y\'\'+2xy\'-12y=0

Let y = x^r be the soltuion of the equation:

y\'= rx^r-1

y\"= r(r-1)x^(r-2)

So, on substitution we get :

x^2r(r-1)x^r-2 +2xr*x^r-1 -12x^r =0

x^r ( r(r-1) + 2r -12) =0

Solve r(r-1) +2r -12 =0

r^2 +r-12 =0

r^2 +4r -3r -12 =0

r( r+4) -3(r +4) =0

(r-3)(r+4) =0

r = 3, -4

GenralSolution : y= c1e^^3x + c2e^-4x

2. x^2y\'\'+xy\'=0

Let y = x^r be a solution

So, y\'= rx^r-1 ; y\" = r(r-1)x^(r-2)

substituting the above:

x^2r(r-1)x^(r-2) + x*rx^(r-1) =0

x^r( r(r-1) + r ) =0

Solve equation : r(r-1) +r =0

r^2 =0 ---> r=0

y = C1

3. x2y\'\'+xy\'-y=0

Let y = x^r be the soltuion of the equation:

y\'= rx^r-1

y\"= r(r-1)x^(r-2)

So, on substitution we get :

x^2 r(r-1)x^(r-2) +xrx^(r-1) - x^r =0

x^r { r(r- 1) +r -1 ) =0

r^2 -r +r -1 =0

solve th equation : r^2 -1 =0

r =1, r=-1

So, general solution : y = c1e^x + c2e^-x