Internal problem ID [10867]
Internal file name [OUTPUT/10124_Sunday_December_24_2023_05_12_44_PM_40460358/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 44.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_y"
Maple gives the following as the ode type
[[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }=0} \]
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right )+a \,x^{n} p \left (x \right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= -a \,x^{n} p \end {align*}
Where \(f(x)=-a \,x^{n}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= -a \,x^{n} \,d x\\ \int { \frac {1}{p} \,dp} &= \int {-a \,x^{n} \,d x}\\ \ln \left (p \right )&=-\frac {a \,x^{n +1}}{n +1}+c_{1}\\ p&={\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}+c_{1}}\\ &=c_{1} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \end {align*}
Since \(p=y^{\prime }\) then the new first order ode to solve is \begin {align*} y^{\prime } = c_{1} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \end {align*}
Integrating both sides gives \begin {align*} y &= \int { c_{1} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}}\,\mathop {\mathrm {d}x}}\\ &= \frac {c_{1} \left (\frac {a}{n +1}\right )^{-\frac {1}{n +1}} \left (\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (\frac {x^{n +1} a \,n^{2}}{n +1}+\frac {2 x^{n +1} a n}{n +1}+n^{2}+\frac {a \,x^{n +1}}{n +1}+3 n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (n +2\right ) \left (2 n +3\right ) a}+\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}+1, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (2 n +3\right ) a}\right )}{n +1}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} \left (\frac {a}{n +1}\right )^{-\frac {1}{n +1}} \left (\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (\frac {x^{n +1} a \,n^{2}}{n +1}+\frac {2 x^{n +1} a n}{n +1}+n^{2}+\frac {a \,x^{n +1}}{n +1}+3 n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (n +2\right ) \left (2 n +3\right ) a}+\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}+1, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (2 n +3\right ) a}\right )}{n +1}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} \left (\frac {a}{n +1}\right )^{-\frac {1}{n +1}} \left (\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (\frac {x^{n +1} a \,n^{2}}{n +1}+\frac {2 x^{n +1} a n}{n +1}+n^{2}+\frac {a \,x^{n +1}}{n +1}+3 n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (n +2\right ) \left (2 n +3\right ) a}+\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}+1, \frac {n +2}{2 n +2}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (2 n +3\right ) a}\right )}{n +1}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+a \,x^{n} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )+a \,x^{n} u \left (x \right )=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & u^{\prime }\left (x \right )=-a \,x^{n} u \left (x \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {u^{\prime }\left (x \right )}{u \left (x \right )}=-a \,x^{n} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {u^{\prime }\left (x \right )}{u \left (x \right )}d x =\int -a \,x^{n}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (u \left (x \right )\right )=-\frac {a \,x^{n +1}}{n +1}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-\frac {-c_{1} n +a \,x^{n +1}-c_{1}}{n +1}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (x \right ) \\ {} & {} & u \left (x \right )={\mathrm e}^{-\frac {-c_{1} n +a \,x^{n +1}-c_{1}}{n +1}} \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} u =y^{\prime } \\ {} & {} & y^{\prime }={\mathrm e}^{-\frac {-c_{1} n +a \,x^{n +1}-c_{1}}{n +1}} \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} y \\ {} & {} & \int y^{\prime }d x =\int {\mathrm e}^{-\frac {-c_{1} n +a \,x^{n +1}-c_{1}}{n +1}}d x +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{-\frac {-c_{1} n -c_{1}}{n +1}} \left (\frac {a}{n +1}\right )^{-\frac {1}{n +1}} \left (\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (\frac {x^{n +1} a \,n^{2}}{n +1}+\frac {2 x^{n +1} a n}{n +1}+n^{2}+\frac {a \,x^{n +1}}{n +1}+3 n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \mathit {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}, \frac {n +2}{2 \left (n +1\right )}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (n +2\right ) \left (2 n +3\right ) a}+\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-1-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \mathit {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}+1, \frac {n +2}{2 \left (n +1\right )}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (2 n +3\right ) a}\right )}{n +1}+c_{2} \end {array} \]
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 <- LODE missing y successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 244
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-n} \left (\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +2\right )^{2} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +1\right )^{3} \left (x^{n} a x +n +2\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+2 c_{1} \left (n +2\right ) a \left (n +\frac {3}{2}\right ) x^{n}\right )}{\left (n +2\right ) \left (2 n +3\right ) a} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 56
DSolve[y''[x]+a*x^n*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2-\frac {c_1 x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {a x^{n+1}}{n+1}\right )}{n+1} \]