Internal problem ID [14051]
Internal file name [OUTPUT/13732_Friday_March_01_2024_09_08_52_AM_63959366/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {-y^{\prime }=1-2 x} \]
Integrating both sides gives \begin {align*} y &= \int { -1+2 x\,\mathop {\mathrm {d}x}}\\ &= \left (x -1\right ) x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (x -1\right ) x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = \left (x -1\right ) x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{\prime }=1-2 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=-1+2 x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-1+2 x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x^{2}+c_{1} -x \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x^{2}+c_{1} -x \end {array} \]
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((2*x-1)-diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{2}+c_{1} -x \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 7
DSolve[(2*x-1)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \]