2.1.14.1 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

\[\begin {aligned} y^{\prime } c&=a x \end {aligned}\]

Because the ODE has the form \(y^{\prime }=f(x)\), the solution requires only integration. Therefore

\begin{align*} dy &= \left (\frac {x a}{c}\right ) \, dx\\ y &= \int { \left (\frac {x a}{c}\right ) \, dx}\\ &= \frac {x^{2} a}{2 c}+c_1 \end{align*}

Summary of solutions found

\[ y = \frac {x^{2} a}{2 c}+c_1 \]