Internal problem ID [12615]
Internal file name [OUTPUT/11268_Friday_November_03_2023_06_29_32_AM_784968/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.1, page 40
Problem number: 1 (B).
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]
\[ \boxed {y^{\prime }=x -1} \]
Integrating both sides gives \begin {align*} y &= \int { x -1\,\mathop {\mathrm {d}x}}\\ &= \frac {x \left (x -2\right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \left (x -2\right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \left (x -2\right )}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=x -1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (x -1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {1}{2} x^{2}-x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {1}{2} x^{2}-x +c_{1} \end {array} \]
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: 14
dsolve(diff(y(x),x)=x-1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{2} x^{2}-x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 18
DSolve[y'[x]==x-1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2}{2}-x+c_1 \]