Internal problem ID [7330]
Internal file name [OUTPUT/6311_Sunday_June_05_2022_04_39_32_PM_74264084/index.tex]
Book: First order enumerated odes
Section: section 1
Problem number: 14.
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]
\[ \boxed {y^{\prime } c=a x} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {x a}{c}\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{2} a}{2 c}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2} a}{2 c}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2} a}{2 c}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } c =a x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x a}{c} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {x a}{c}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{2} a}{2 c}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {a \,x^{2}+2 c_{1} c}{2 c} \end {array} \]
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: 15
dsolve(c*diff(y(x),x)=a*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {a \,x^{2}}{2 c}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 19
DSolve[c*y'[x]==a*x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {a x^2}{2 c}+c_1 \]