Internal problem ID [9137]
Internal file name [OUTPUT/8072_Monday_June_06_2022_01_39_38_AM_91568443/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 803.
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.
\[ \boxed {y^{\prime }-\frac {f_{1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime } \sqrt {y^{2}}\, x +f_{1} \left (y^{2}-2 \ln \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f_{1} \left (y^{2}-2 \ln \left (x \right )\right )}{\sqrt {y^{2}}\, x} \end {array} \]
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 `, `-> Computing symmetries using: way = 5`[0, (f__1(y^2-2*ln(x))*(y^2)^(1/2)-y)/y^2], [0, (f__1(y^2-2*ln(x))*y-(y^2)^(1/2
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 65
dsolve(diff(y(x),x) = _F1(y(x)^2-2*ln(x))/(y(x)^2)^(1/2)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} \right )+c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {2 \ln \left (x \right )+2 \operatorname {RootOf}\left (\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (2 \textit {\_a} \right )-1}d \textit {\_a} \right )+c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.128 (sec). Leaf size: 603
DSolve[y'[x] == F1[-2*Log[x] + y[x]^2]/(x*Sqrt[y[x]^2]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]^2} \text {F1}\left (K[2]^2-2 \log (x)\right )}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}+\frac {K[2]}{\left (\text {F1}\left (K[2]^2-2 \log (x)\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (x)\right )+1\right )}-\int _1^x\left (\frac {2 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {4 K[2] \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right )^2 \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}+\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right ) \text {F1}\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right )^2 K[1]}-\frac {2 \sqrt {K[2]^2} \text {F1}'\left (K[2]^2-2 \log (K[1])\right )}{\left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (K[2]^2-2 \log (K[1])\right )+1\right ) K[1]}\right )dK[1]\right )dK[2]+\int _1^x\left (-\frac {\text {F1}\left (y(x)^2-2 \log (K[1])\right )^2}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1]}-\frac {\sqrt {y(x)^2} \text {F1}\left (y(x)^2-2 \log (K[1])\right )}{\left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )-1\right ) \left (\text {F1}\left (y(x)^2-2 \log (K[1])\right )+1\right ) K[1] y(x)}\right )dK[1]=c_1,y(x)\right ] \]