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DEtools[ode_int_y] - given the nth order linear ODE satisfied by y(x), compute the nth order linear ODE satisfied by int(y(x),x)
DEtools[ode_y1] - given the nth order linear ODE satisfied by y(x), compute the nth order linear ODE satisfied by diff(y(x),x)
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Calling Sequence
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ode_int_y(ode, y(x))
ode_y1(ode, y(x))
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Parameters
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ode
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ordinary differential equation satisfied by y(x)
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y(x)
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unknown function of one variable
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Examples
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For enhanced input output use DEtools[diff_table] and PDEtools[declare].
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![[diff_table, ode_int_y, ode_y1]](/support/helpjp/helpview.aspx?si=8671/file01358/math86.png)
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, c(x))](/support/helpjp/helpview.aspx?si=8671/file01358/math90.png)
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| (2) |
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, c(x), prime = x)](/support/helpjp/helpview.aspx?si=8671/file01358/math107.png)
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| (3) |
Now, if  satisfies
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*Y[]+c[1](x)*Y[x]+c[2](x)*Y[x, x]+Y[x, x, x, x] = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math124.png)
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![c[0]*y+c[1]*`y'`+c[2]*`y''`+`y''''` = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math127.png)
| (4) |
then the derivative of  satisfies
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 = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math135.png)
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![`y''''`-c[0]^`'`*`y'''`/c[0]+c[2]*`y''`-(-c[2]^`'`*c[0]+c[0]^`'`*c[2]-c[1]*c[0])*`y'`/c[0]-(-c[1]^`'`*c[0]+c[0]^`'`*c[1]-c[0]^2)*y/c[0] = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math138.png)
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and so, the integral of the function  in the equation above satisfy this other ODE (the starting point)
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*`y'`/c[0]-(-c[1]^`'`*c[0]+c[0]^`'`*c[1]-c[0]^2)*y/c[0] = 0, y(x)) = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math146.png)
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![c[0]*y+c[1]*`y'`+c[2]*`y''`+`y''''` = 0](/support/helpjp/helpview.aspx?si=8671/file01358/math149.png)
| (6) |
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