Internal problem ID [3966]
Internal file name [OUTPUT/3459_Sunday_June_05_2022_09_21_25_AM_50879331/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 720.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Bernoulli]
Unable to solve or complete the solution.
\[ \boxed {f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n}}{f \left (x \right ) y^{m}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 61
dsolve(f(x)*y(x)^m*diff(y(x),x)+g(x)*y(x)^(m+1)+h(x)*y(x)^n = 0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )} {\left (\left (n -m -1\right ) \left (\int \frac {h \left (x \right ) {\mathrm e}^{\left (-n +m +1\right ) \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )}}{f \left (x \right )}d x \right )+c_{1} \right )}^{\frac {1}{-n +m +1}} \]
✓ Solution by Mathematica
Time used: 14.159 (sec). Leaf size: 187
DSolve[f[x] y[x]^m y'[x]+ g[x] y[x]^(m+1)+ h[x] y[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\exp \left ((m-n+1) \int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \left ((m-n+1) \int _1^x-\frac {\exp \left (-\left ((m-n+1) \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right )\right ) h(K[2])}{f(K[2])}dK[2]+c_1\right )\right ){}^{\frac {1}{m-n+1}} \\ y(x)\to \left ((m-n+1) \exp \left ((m-n+1) \int _1^x-\frac {g(K[1])}{f(K[1])}dK[1]\right ) \int _1^x-\frac {\exp \left (-\left ((m-n+1) \int _1^{K[2]}-\frac {g(K[1])}{f(K[1])}dK[1]\right )\right ) h(K[2])}{f(K[2])}dK[2]\right ){}^{\frac {1}{m-n+1}} \\ \end{align*}