Internal problem ID [7113]
Internal file name [OUTPUT/6099_Sunday_June_05_2022_04_21_48_PM_25779835/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 69.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {a y^{2} y^{\prime \prime }+b y^{2}=c} \]
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*} a \,y^{2} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+b \,y^{2} = c \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 {b \,y^{2}-c}{a \,y^{2} p} \end {align*}
Where \(f(y)=-\frac {b \,y^{2}-c}{a \,y^{2}}\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= -\frac {b \,y^{2}-c}{a \,y^{2}} \,d y \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {-\frac {b \,y^{2}-c}{a \,y^{2}} \,d y} \\ \frac {p^{2}}{2}&=-\frac {b y +\frac {c}{y}}{a}+c_{1} \\ \end{align*} The solution is \[ \frac {p \left (y \right )^{2}}{2}+\frac {b y +\frac {c}{y}}{a}-c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {{y^{\prime }}^{2}}{2}+\frac {b y+\frac {c}{y}}{a}-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 {-2 a y \left (b y^{2}-c_{1} a y+c \right )}}{a y} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {-2 a y \left (b y^{2}-c_{1} a y+c \right )}}{a y} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {a y}{\sqrt {-2 a y \left (-a c_{1} y +b \,y^{2}+c \right )}}d y &= \int {dx}\\ \int _{}^{y}\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a}&= x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {a y}{\sqrt {-2 a y \left (-a c_{1} y +b \,y^{2}+c \right )}}d y &= \int {dx}\\ \int _{}^{y}-\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a}&= x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a} &= x +c_{2} \\ \tag{2} \int _{}^{y}-\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a} = x +c_{2} \] Verified OK.
\[ \int _{}^{y}-\frac {a \textit {\_a}}{\sqrt {-2 a \textit {\_a} \left (\textit {\_a}^{2} b -\textit {\_a} a c_{1} +c \right )}}d \textit {\_a} = x +c_{3} \] 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)+(_a^2*b-c)/(_a^2*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: 0.062 (sec). Leaf size: 76
dsolve(a*y(x)^2*diff(y(x),x$2)+b*y(x)^2=c,y(x), singsol=all)
\begin{align*} a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} a \left (-2 b \,\textit {\_a}^{2}+\textit {\_a} a c_{1} -2 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}}{\sqrt {\textit {\_a} a \left (-2 b \,\textit {\_a}^{2}+\textit {\_a} a c_{1} -2 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.801 (sec). Leaf size: 346
DSolve[a*y[x]^2*y''[x]+b*y[x]^2==c,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\left (\sqrt {-16 b c+a^2 c_1{}^2}-a c_1\right ) \left (\sqrt {-16 b c+a^2 c_1{}^2}+a c_1\right ){}^2 \left (1+\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}-a c_1}\right ) \left (1-\frac {4 b y(x)}{\sqrt {-16 b c+a^2 c_1{}^2}+a c_1}\right ) \left (E\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right )|\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (2 \sqrt {\frac {b}{\sqrt {a^2 c_1{}^2-16 b c}-a c_1}} \sqrt {y(x)}\right ),\frac {a c_1-\sqrt {a^2 c_1{}^2-16 b c}}{a c_1+\sqrt {a^2 c_1{}^2-16 b c}}\right )\right ){}^2}{16 b^3 y(x) \left (-\frac {2 \left (b y(x)^2+c\right )}{a y(x)}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \]