Internal problem ID [9922]
Internal file name [OUTPUT/8869_Monday_June_06_2022_05_42_49_AM_22601388/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1600 (6.10).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x", "second_order_ode_can_be_made_integrable"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {y^{\prime \prime }+y^{2} b +c y+a y^{3}=-d} \]
Multiplying the ode by \(y^{\prime }\) gives \[ y^{\prime } y^{\prime \prime }+\left (b y+c +a y^{2}\right ) y^{\prime } y+y^{\prime } d = 0 \] Integrating the above w.r.t \(x\) gives \begin {align*} \int \left (y^{\prime } y^{\prime \prime }+\left (b y+c +a y^{2}\right ) y^{\prime } y+y^{\prime } d \right )d x &= 0 \\ d y+\frac {{y^{\prime }}^{2}}{2}+\frac {a y^{4}}{4}+\frac {b y^{3}}{3}+\frac {c y^{2}}{2} = c_2 \end {align*}
Which is now solved for \(y\). Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {-18 a y^{4}-24 b y^{3}-36 c y^{2}-72 d y+72 c_{1}}}{6} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {-18 a y^{4}-24 b y^{3}-36 c y^{2}-72 d y+72 c_{1}}}{6} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (a \,y^{2}+b y +c \right ) y = -d \end {align*}
Which is now solved as first order ode for \(p(y)\). In canonical form the ODE is \begin {align*} p' &= F(y,p)\\ &= f( y) g(p)\\ &= \frac {-a \,y^{3}-y^{2} b -c y -d}{p} \end {align*}
Where \(f(y)=-a \,y^{3}-y^{2} b -c y -d\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= -a \,y^{3}-y^{2} b -c y -d \,d y \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {-a \,y^{3}-y^{2} b -c y -d \,d y} \\ \frac {p^{2}}{2}&=-\frac {1}{4} a \,y^{4}-\frac {1}{3} b \,y^{3}-\frac {1}{2} c \,y^{2}-d y +c_{1} \\ \end{align*} The solution is \[ \frac {p \left (y \right )^{2}}{2}+\frac {a \,y^{4}}{4}+\frac {b \,y^{3}}{3}+\frac {c \,y^{2}}{2}+d y -c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} d y+\frac {{y^{\prime }}^{2}}{2}+\frac {a y^{4}}{4}+\frac {b y^{3}}{3}+\frac {c y^{2}}{2}-c_{1} = 0 \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {-18 a y^{4}-24 b y^{3}-36 c y^{2}-72 d y+72 c_{1}}}{6} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {-18 a y^{4}-24 b y^{3}-36 c y^{2}-72 d y+72 c_{1}}}{6} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+d+b*_a^2+_a*c+_a^3*a = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.453 (sec). Leaf size: 89
dsolve(diff(diff(y(x),x),x)+d+y(x)^2*b+c*y(x)+a*y(x)^3=0,y(x), singsol=all)
\begin{align*} -6 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 \textit {\_a}^{2} c -72 \textit {\_a} d +36 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 6 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 \textit {\_a}^{2} c -72 \textit {\_a} d +36 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.11 (sec). Leaf size: 1017
DSolve[d + c*y[x] + b*y[x]^2 + a*y[x]^3 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right ) \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]\right )}{\left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right ) \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]\right )}}\right ),\frac {\left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,3\right ]\right ) \left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right )}{\left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,3\right ]\right ) \left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right )}\right ){}^2 \left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]\right ){}^2 \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]\right ) \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]\right ) \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,3\right ]\right ) \left (y(x)-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right )}{\left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]\right ){}^2 \left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,1\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,3\right ]\right ) \left (\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,2\right ]-\text {Root}\left [3 a \text {$\#$1}^4+4 b \text {$\#$1}^3+6 c \text {$\#$1}^2+12 d \text {$\#$1}-6 c_1\&,4\right ]\right ) \left (-\frac {1}{2} a y(x)^4-\frac {2}{3} b y(x)^3-c y(x)^2-2 d y(x)+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]