Internal problem ID [14947]
Internal file name [OUTPUT/14957_Monday_April_15_2024_12_04_20_AM_5802521/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: 21.
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: 0.016 (sec). Leaf size: 12
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: 0.002 (sec). Leaf size: 16
DSolve[y'[x]==x+1,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2}{2}+x+c_1 \]