1.25 problem 25

Internal problem ID [5738]
Internal file name [OUTPUT/4986_Sunday_June_05_2022_03_15_59_PM_26046726/index.tex]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number: 25.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {x^{\prime }=-t +1} \]

1.25.1 Solving as separable ode

In canonical form the ODE is \begin {align*} x' &= F(t,x)\\ &= f( t) g(x)\\ &= -t +1 \end {align*}

Where \(f(t)=-t +1\) and \(g(x)=1\). Integrating both sides gives \begin{align*} \frac {1}{1} \,dx &= -t +1 \,d t \\ \int { \frac {1}{1} \,dx} &= \int {-t +1 \,d t} \\ x&=-\frac {1}{2} t^{2}+t +c_{1} \\ \end{align*} Which results in \begin{align*} x &= -\frac {1}{2} t^{2}+t +c_{1} \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= -\frac {1}{2} t^{2}+t +c_{1} \\ \end{align*}

Figure 20: Slope field plot

Verification of solutions

\[ x = -\frac {1}{2} t^{2}+t +c_{1} \] Verified OK.

1.25.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }=-t +1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int x^{\prime }d t =\int \left (-t +1\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=-\frac {1}{2} t^{2}+t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=-\frac {1}{2} t^{2}+t +c_{1} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 12

dsolve(diff(x(t),t)+t=1,x(t), singsol=all)
 

\[ x \left (t \right ) = -\frac {1}{2} t^{2}+t +c_{1} \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 16

DSolve[x'[t]+t==1,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to -\frac {t^2}{2}+t+c_1 \]