Solve the homogeneous differential equation with functional coefficients
I have to solve this equation:
$$ y''-frac{1}{x} y' + frac{1}{x^3} y =0 $$
This is a nonconstant coefficient homogeneous differential equation.
How should I do it?
I tried with the supstitution: $y(x)=u(x)sqrt{x}$, and then I got this equation:
$$u''+(frac{1}{x^3}-frac{3}{4x^2})u=0$$, which seems to me easier to solve, but I don't know how to solve this either!
If someone could help me, I would be very grateful.
differential-equations
asked 2 days ago
stakindmidl
667
add a comment |
I have to solve this equation:
$$ y''-frac{1}{x} y' + frac{1}{x^3} y =0 $$
This is a nonconstant coefficient homogeneous differential equation.
How should I do it?
I tried with the supstitution: $y(x)=u(x)sqrt{x}$, and then I got this equation:
$$u''+(frac{1}{x^3}-frac{3}{4x^2})u=0$$, which seems to me easier to solve, but I don't know how to solve this either!
If someone could help me, I would be very grateful.
differential-equations
asked 2 days ago
stakindmidl
667
1
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago
add a comment |
I have to solve this equation:
$$ y''-frac{1}{x} y' + frac{1}{x^3} y =0 $$
This is a nonconstant coefficient homogeneous differential equation.
How should I do it?
I tried with the supstitution: $y(x)=u(x)sqrt{x}$, and then I got this equation:
$$u''+(frac{1}{x^3}-frac{3}{4x^2})u=0$$, which seems to me easier to solve, but I don't know how to solve this either!
If someone could help me, I would be very grateful.
differential-equations
asked 2 days ago
stakindmidl
667
I have to solve this equation:
$$ y''-frac{1}{x} y' + frac{1}{x^3} y =0 $$
This is a nonconstant coefficient homogeneous differential equation.
How should I do it?
I tried with the supstitution: $y(x)=u(x)sqrt{x}$, and then I got this equation:
$$u''+(frac{1}{x^3}-frac{3}{4x^2})u=0$$, which seems to me easier to solve, but I don't know how to solve this either!
If someone could help me, I would be very grateful.
differential-equations
differential-equations
asked 2 days ago
stakindmidl
667
asked 2 days ago
stakindmidl
667
asked 2 days ago
stakindmidl
667
asked 2 days ago
stakindmidl
667
asked 2 days ago
stakindmidl
667
667
1
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago
add a comment |
1
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago
1
1
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago
add a comment |
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1
Do you have any reason to believe that a simple solution exists? WolframAlpha only finds a solution using Bessel functions. Now, WA is not always simplifying perfectly, so it's possible the solution is simpler than what it says.
– Arthur
2 days ago
The solution containes the Bessel-function:$$y left( x right) ={it _C1},x{{ J}_{2}left(2,{frac {1}{ sqrt {x}}}right)}+{it _C2},x{{ Y}_{2}left(2,{frac {1}{ sqrt {x}}}right)}$$
– Dr. Sonnhard Graubner
2 days ago