Internal problem ID [8928]
Internal file name [OUTPUT/7863_Monday_June_06_2022_12_48_22_AM_47801249/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 594.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {F \left (-\frac {-y^{2}+b}{x^{2}}\right ) x}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-F \left (\frac {y^{2}-b}{x^{2}}\right ) x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {F \left (\frac {y^{2}-b}{x^{2}}\right ) x}{y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[x, -(-y^2+b)/y]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 95
dsolve(diff(y(x),x) = F(-(-y(x)^2+b)/x^2)*x/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-F \left (\frac {\textit {\_Z}^{2}-b}{x^{2}}\right ) x^{2}+\textit {\_Z}^{2}-b \right ) \\ y \left (x \right ) &= \sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} +2 c_{1} \right ) x^{2}+b} \\ y \left (x \right ) &= -\sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} +2 c_{1} \right ) x^{2}+b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.428 (sec). Leaf size: 236
DSolve[y'[x] == (x*F[(-b + y[x]^2)/x^2])/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{-F\left (\frac {K[2]^2-b}{x^2}\right ) x^2+K[2]^2-b}-\int _1^x\left (\frac {F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1] \left (2 K[2] F'\left (\frac {K[2]^2-b}{K[1]^2}\right )-2 K[2]\right )}{\left (F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1]^2-K[2]^2+b\right )^2}-\frac {2 K[2] F'\left (\frac {K[2]^2-b}{K[1]^2}\right )}{K[1] \left (F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1]^2-K[2]^2+b\right )}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F\left (\frac {y(x)^2-b}{K[1]^2}\right ) K[1]}{F\left (\frac {y(x)^2-b}{K[1]^2}\right ) K[1]^2-y(x)^2+b}dK[1]=c_1,y(x)\right ] \]