Internal problem ID [11402]
Internal file name [OUTPUT/10385_Wednesday_May_17_2023_08_10_24_PM_21188209/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises
page 41
Problem number: 1(e).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {x x^{\prime }+x t=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x x^{\prime }+x t =1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=\frac {1-x t}{x} \end {array} \]
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 differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
dsolve(x(t)*diff(x(t),t)=1-t*x(t),x(t), singsol=all)
\[ x \left (t \right ) = -\frac {\left (2^{\frac {2}{3}} t^{2}-4 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 2^{\frac {1}{3}} c_{1} t +2^{\frac {1}{3}} t \operatorname {AiryAi}\left (\textit {\_Z} \right )-2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} -2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right )\right ) 2^{\frac {1}{3}}}{4} \]
✓ Solution by Mathematica
Time used: 0.399 (sec). Leaf size: 121
DSolve[x[t]*x'[t]==1-t*x[t],x[t],t,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {(-1)^{2/3} \sqrt [3]{2} t \operatorname {AiryAi}\left (-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} \left (t^2+2 x(t)\right )\right )-2 \operatorname {AiryAiPrime}\left (-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} \left (t^2+2 x(t)\right )\right )}{(-1)^{2/3} \sqrt [3]{2} t \operatorname {AiryBi}\left (-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} \left (t^2+2 x(t)\right )\right )-2 \operatorname {AiryBiPrime}\left (-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} \left (t^2+2 x(t)\right )\right )}+c_1=0,x(t)\right ] \]