5.6 problem 1(f)

Internal problem ID [11403]
Internal file name [OUTPUT/10386_Wednesday_May_17_2023_08_10_25_PM_15906252/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(f).
ODE order: 1.
ODE degree: 2.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[`y=_G(x,y')`]

Unable to solve or complete the solution.

\[ \boxed {{x^{\prime }}^{2}+x t=\sqrt {1+t}} \] Solving the given ode for \(x^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} x^{\prime }&=\sqrt {-x t +\sqrt {1+t}} \tag {1} \\ x^{\prime }&=-\sqrt {-x t +\sqrt {1+t}} \tag {2} \end {align*}

Now each one of the above ODE is solved.

Solving equation (1)

Unable to determine ODE type.

Unable to determine ODE type.

Solving equation (2)

Unable to determine ODE type.

Unable to determine ODE type.

5.6.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {x^{\prime }}^{2}+x t =\sqrt {1+t} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [x^{\prime }=\sqrt {-x t +\sqrt {1+t}}, x^{\prime }=-\sqrt {-x t +\sqrt {1+t}}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} x^{\prime }=\sqrt {-x t +\sqrt {1+t}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} x^{\prime }=-\sqrt {-x t +\sqrt {1+t}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
trying 1st order WeierstrassPPrime solution for high degree ODE 
trying 1st order JacobiSN solution for high degree ODE 
trying 1st order ODE linearizable_by_differentiation 
trying differential order: 1; missing variables 
trying dAlembert 
trying simple symmetries for implicit equations 
Successful isolation of dx/dt: 2 solutions were found. Trying to solve each resulting ODE. 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   trying exact 
   Looking for potential symmetries 
   trying an equivalence to an Abel ODE 
   trying 1st order ODE linearizable_by_differentiation 
-> Solving 1st order ODE of high degree, Lie methods, 1st trial 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 2 
`, `-> Computing symmetries using: way = 2 
-> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods 
-> Calling odsolve with the ODE`, diff(y(x), x) = -2*y(x)^2*x^2*(1+(-4*y(x)*x^2+4*y(x)^2+1)^(1/2))/(-y(x)*(-4*y(x)*x^2+4*y(x)^2+1)^( 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying homogeneous types: 
   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(y(x), x) = -4*y(x)*(y(x)+1)^(1/2)*x/(2*x*(y(x)+1)^(1/2)*y(x)^2-2*(y(x)+1)^(1/2)*x^2+y(x)+2), 
   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 
   trying exact 
   Looking for potential symmetries 
   trying inverse_Riccati 
   trying an equivalence to an Abel ODE 
   differential order: 1; trying a linearization to 2nd order 
   --- trying a change of variables {x -> y(x), y(x) -> x} 
   differential order: 1; trying a linearization to 2nd order 
   trying 1st order ODE linearizable_by_differentiation 
-> Solving 1st order ODE of high degree, Lie methods, 2nd trial 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 5 
`, `-> Computing symmetries using: way = 5`
 

✗ Solution by Maple

dsolve(diff(x(t),t)^2+t*x(t)=sqrt(1+t),x(t), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[x'[t]^2+t*x[t]==Sqrt[1+t],x[t],t,IncludeSingularSolutions -> True]
 

Not solved