Internal problem ID [9457]
Internal file name [OUTPUT/8397_Monday_June_06_2022_02_56_42_AM_78713603/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1128.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_change_of_variable_on_y_method_2"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (x f \left (x \right )+2\right ) y^{\prime }+f \left (x \right ) y=0} \]
In normal form the ode \begin {align*} x y^{\prime \prime }+\left (x f \left (x \right )+2\right ) y^{\prime }+f \left (x \right ) y&=0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {x f \left (x \right )+2}{x}\\ q \left (x \right )&=\frac {f \left (x \right )}{x} \end {align*}
Applying change of variables on the depndent variable \(y = v \left (x \right ) x^{n}\) to (2) gives the following ode where the dependent variables is \(v \left (x \right )\) and not \(y\). \begin {align*} v^{\prime \prime }\left (x \right )+\left (\frac {2 n}{x}+p \right ) v^{\prime }\left (x \right )+\left (\frac {n \left (n -1\right )}{x^{2}}+\frac {n p}{x}+q \right ) v \left (x \right )&=0 \tag {3} \end {align*}
Let the coefficient of \(v \left (x \right )\) above be zero. Hence \begin {align*} \frac {n \left (n -1\right )}{x^{2}}+\frac {n p}{x}+q&=0 \tag {4} \end {align*}
Substituting the earlier values found for \(p \left (x \right )\) and \(q \left (x \right )\) into (4) gives \begin {align*} \frac {n \left (n -1\right )}{x^{2}}+\frac {n \left (x f \left (x \right )+2\right )}{x^{2}}+\frac {f \left (x \right )}{x}&=0 \tag {5} \end {align*}
Solving (5) for \(n\) gives \begin {align*} n&=-1 \tag {6} \end {align*}
Substituting this value in (3) gives \begin {align*} v^{\prime \prime }\left (x \right )+\left (-\frac {2}{x}+\frac {x f \left (x \right )+2}{x}\right ) v^{\prime }\left (x \right )&=0 \\ v^{\prime \prime }\left (x \right )+f \left (x \right ) v^{\prime }\left (x \right )&=0 \tag {7} \\ \end {align*}
Using the substitution \begin {align*} u \left (x \right ) = v^{\prime }\left (x \right ) \end {align*}
Then (7) becomes \begin {align*} u^{\prime }\left (x \right )+f \left (x \right ) u \left (x \right ) = 0 \tag {8} \\ \end {align*}
The above is now solved for \(u \left (x \right )\). In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -f \left (x \right ) u \end {align*}
Where \(f(x)=-f \left (x \right )\) and \(g(u)=u\). Integrating both sides gives \begin {align*} \frac {1}{u} \,du &= -f \left (x \right ) \,d x\\ \int { \frac {1}{u} \,du} &= \int {-f \left (x \right ) \,d x}\\ \ln \left (u \right )&=\int -f \left (x \right )d x +c_{1}\\ u&={\mathrm e}^{\int -f \left (x \right )d x +c_{1}}\\ &=c_{1} {\mathrm e}^{\int -f \left (x \right )d x} \end {align*}
Now that \(u \left (x \right )\) is known, then \begin {align*} v^{\prime }\left (x \right )&= u \left (x \right )\\ v \left (x \right )&= \int u \left (x \right )d x +c_{2}\\ &= \int c_{1} {\mathrm e}^{\int -f \left (x \right )d x}d x +c_{2} \end {align*}
Hence \begin {align*} y&= v \left (x \right ) x^{n}\\ &= \frac {\int c_{1} {\mathrm e}^{\int -f \left (x \right )d x}d x +c_{2}}{x}\\ &= \frac {c_{1} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )+c_{2}}{x}\\ \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\int c_{1} {\mathrm e}^{\int -f \left (x \right )d x}d x +c_{2}}{x} \\ \end{align*}
Verification of solutions
\[ y = \frac {\int c_{1} {\mathrm e}^{\int -f \left (x \right )d x}d x +c_{2}}{x} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying differential order: 2; exact nonlinear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> trying reduction of order to Riccati trying Riccati sub-methods: trying Riccati_symmetries -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[0, y]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 33
dsolve(x*diff(diff(y(x),x),x)+(x*f(x)+2)*diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{2} \left (\int {\mathrm e}^{-\left (\int \frac {f \left (x \right ) x +2}{x}d x \right )} x^{2}d x \right )+c_{1}}{x} \]
✓ Solution by Mathematica
Time used: 0.195 (sec). Leaf size: 37
DSolve[f[x]*y[x] + (2 + x*f[x])*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_2 \int _1^x\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right )dK[2]+c_1}{x} \]