2.1.14.4 ✓ Maple. Time used: 0.001 (sec). Leaf size: 15

ode:=c*diff(y(x),x) = a*x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = \frac {a \,x^{2}}{2 c}+c_1 \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & c \left (\frac {d}{d x}y \left (x \right )\right )=a x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a x}{c} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int \frac {a x}{c}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\frac {a \,x^{2}}{2 c}+\mathit {C1} \end {array} \]