Internal problem ID [3390]
Internal file name [OUTPUT/2883_Sunday_June_05_2022_08_45_08_AM_37908114/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 133.
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.
\[ \boxed {y^{\prime }-x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x^{m -1} y^{1-n} f \left (a \,x^{m}+b y^{n}\right ) \end {array} \]
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)*a*m*x^m/(b*y(x)^n*n*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: 0.5 (sec). Leaf size: 174
dsolve(diff(y(x),x) = x^(m-1)*y(x)^(1-n)*f(a*x^m+b*y(x)^n),y(x), singsol=all)
\[ y \left (x \right ) = {\left (-\frac {-\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} b n \textit {\_a} -\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} a m n +a \,m^{2}}d \textit {\_a} \right ) b \,m^{2}+c_{1} m -x^{m}\right ) b +a \,x^{m}}{b}\right )}^{\frac {1}{n}} \]
✓ Solution by Mathematica
Time used: 0.509 (sec). Leaf size: 242
DSolve[y'[x]==x^(m-1) y[x]^(1-n) f[a x^m + b y[x]^n],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {a m K[2]^{n-1}}{a m+b n f\left (a x^m+b K[2]^n\right )}-\int _1^x\left (\frac {a b m n K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{a m+b n f\left (a K[1]^m+b K[2]^n\right )}-\frac {a b^2 m n^2 f\left (a K[1]^m+b K[2]^n\right ) K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{\left (a m+b n f\left (a K[1]^m+b K[2]^n\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {a m f\left (a K[1]^m+b y(x)^n\right ) K[1]^{m-1}}{a m+b n f\left (a K[1]^m+b y(x)^n\right )}dK[1]=c_1,y(x)\right ] \]