Internal problem ID [7051]
Internal file name [OUTPUT/6037_Sunday_June_05_2022_04_14_51_PM_54705140/index.tex]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 7.
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]
\[ \boxed {y^{\prime }=x} \]
Integrating both sides gives \begin {align*} y &= \int { x\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{2}}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2}}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2}}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x^{2}}{2}+c_{1} \end {array} \]
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: 11
dsolve(diff(y(x),x) = x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{2}}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 15
DSolve[y'[x] == x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^2}{2}+c_1 \]