Internal problem ID [6119]
Internal file name [OUTPUT/5367_Sunday_June_05_2022_03_35_33_PM_12539902/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS.
Page 9
Problem number: 2(a).
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]
\[ \boxed {y^{\prime }={\mathrm e}^{3 x}-x} \]
Integrating both sides gives \begin {align*} y &= \int { {\mathrm e}^{3 x}-x\,\mathop {\mathrm {d}x}}\\ &= -\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{3 x}-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left ({\mathrm e}^{3 x}-x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \end {array} \]
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: 17
dsolve(diff(y(x),x)=exp(3*x)-x,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{2}}{2}+\frac {{\mathrm e}^{3 x}}{3}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 24
DSolve[y'[x]==Exp[3*x]-x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {x^2}{2}+\frac {e^{3 x}}{3}+c_1 \]