Internal problem ID [15053]
Internal file name [OUTPUT/15054_Sunday_April_21_2024_01_21_36_PM_25578689/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page
54
Problem number: 161.
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(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-2 y \,{\mathrm e}^{x}-2 \sqrt {y \,{\mathrm e}^{x}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-2 y \,{\mathrm e}^{x}-2 \sqrt {y \,{\mathrm e}^{x}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=2 y \,{\mathrm e}^{x}+2 \sqrt {y \,{\mathrm e}^{x}} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)+2*y(x)*exp(2*x)-y(x)*exp(x)-3*(diff(y(x), x))*exp(x)-(1/2)*(diff(y(x), x))- Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] trying symmetries linear in x and y(x) -> Try solving first the homogeneous part of the ODE trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Kummer successful <- special function solution successful -> Trying to convert hypergeometric functions to elementary form... <- elementary form is not straightforward to achieve - returning special function solution free of uncomputed integra <- Kovacics algorithm successful Change of variables used: [x = ln(t)] Linear ODE actually solved: (4*t-2)*u(t)+(-6*t+1)*diff(u(t),t)+2*t*diff(diff(u(t),t),t) = 0 <- change of variables successful <- solving first the homogeneous part of the ODE successful --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> 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)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> 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 a symmetry pattern of the form [F(x),G(x)*y+H(x)] <- symmetry pattern of the form [F(x),G(x)*y+H(x)] successful 1st order, trying the canonical coordinates of the invariance group -> Calling odsolve with the ODE`, diff(y(x), x) = 2*y(x)*(exp((1/2)*x))^2, y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful trying an integrating factor from the invariance group <- integrating factor successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 53
dsolve(diff(y(x),x)-2*y(x)*exp(x)=2*sqrt(y(x)*exp(x)),y(x), singsol=all)
\[ \frac {y \left (x \right ) {\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}}-\left (\int {\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}}d x \right ) \sqrt {y \left (x \right ) {\mathrm e}^{x}}+c_{1} \sqrt {y \left (x \right ) {\mathrm e}^{x}}}{\sqrt {y \left (x \right ) {\mathrm e}^{x}}} = 0 \]
✓ Solution by Mathematica
Time used: 0.215 (sec). Leaf size: 56
DSolve[y'[x]-2*y[x]*Exp[x]==2*Sqrt[y[x]*Exp[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 \left (\sqrt {\pi } \sqrt {y(x)} \text {erf}\left (\frac {\sqrt {e^x y(x)}}{\sqrt {y(x)}}\right )-e^{-e^x} y(x)\right )}{\sqrt {y(x)}}=c_1,y(x)\right ] \]