Internal problem ID [8391]
Internal file name [OUTPUT/7324_Sunday_June_05_2022_05_50_05_PM_43412017/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 54.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Chini, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} f \left (x \right )+f \left (x \right )^{2} g^{\prime }\left (x \right )+f^{\prime }\left (x \right ) y-y^{\prime } f \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} f \left (x \right )+f \left (x \right )^{2} g^{\prime }\left (x \right )+f^{\prime }\left (x \right ) y}{f \left (x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini <- Chini successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 38
dsolve(diff(y(x),x) - a^n*f(x)^(1-n)*diff(g(x),x)*y(x)^n - diff(f(x),x)*y(x)/f(x) - f(x)*diff(g(x),x)=0,y(x), singsol=all)
\[ \frac {a y \left (x \right ) \operatorname {LerchPhi}\left (-\left (\frac {a y \left (x \right )}{f \left (x \right )}\right )^{n}, 1, \frac {1}{n}\right )}{n f \left (x \right )}-a g \left (x \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 74
DSolve[y'[x] - a^n*f[x]^(1-n)*g'[x]*y[x]^n - f'[x]*y[x]/f[x] - f[x]*g'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [y(x) \left (a^n f(x)^{-n}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\left (\left (a^n f(x)^{-n}\right )^{\frac {1}{n}} y(x)\right )^n\right )=f(x) g(x) \left (a^n f(x)^{-n}\right )^{\frac {1}{n}}+c_1,y(x)\right ] \]