Internal problem ID [8422]
Internal file name [OUTPUT/7355_Sunday_June_05_2022_10_53_33_PM_79424869/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 85.
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.
\[ \boxed {y^{\prime }-x^{-1+a} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-x^{-1+a} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x^{-1+a} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right ) \end {array} \]
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 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 1st order, trying the canonical coordinates of the invariance group -> Calling odsolve with the ODE`, diff(y(x), x) = -y(x)*x^a/(y(x)^b*x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- 1st order, canonical coordinates successful <- symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] successful`
✓ Solution by Maple
Time used: 0.235 (sec). Leaf size: 152
dsolve(diff(y(x),x) - x^(a-1)*y(x)^(1-b)*f(x^a/a + y(x)^b/b)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\left (\frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} \left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} b +f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} \left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} \textit {\_a} +a}d \textit {\_a} \right ) a^{2}+a b c_{1} -x^{a} b \right ) a -x^{a} b}{a}\right )}^{\frac {1}{b}} \]
✓ Solution by Mathematica
Time used: 0.322 (sec). Leaf size: 238
DSolve[y'[x] - x^(a-1)*y[x]^(1-b)*f[x^a/a + y[x]^b/b]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]^{b-1}}{f\left (\frac {x^a}{a}+\frac {K[2]^b}{b}\right )+1}-\int _1^x\left (\frac {K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1}-\frac {f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right ) K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{\left (f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right ) K[1]^{a-1}}{f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )+1}dK[1]=c_1,y(x)\right ] \]