Internal problem ID [14959]
Internal file name [OUTPUT/14969_Monday_April_15_2024_12_04_30_AM_89943328/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 2. The method of isoclines. Exercises page 27
Problem number: 33.
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]
\[ \boxed {y^{\prime }=1-x} \]
Integrating both sides gives \begin {align*} y &= \int { 1-x\,\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.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=1-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (1-x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x -\frac {1}{2} x^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x -\frac {1}{2} x^{2}+c_{1} \end {array} \]
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(y(x),x)=1-x,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {1}{2} x^{2}+x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 16
DSolve[y'[x]==1-x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {x^2}{2}+x+c_1 \]