1.1 problem 1 (a)

Internal problem ID [5912]
Internal file name [OUTPUT/5160_Sunday_June_05_2022_03_26_32_PM_48421319/index.tex]

Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section: Chapter 1.3 Introduction– Linear equations of First Order. Page 38
Problem number: 1 (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]

\[ \boxed {y^{\prime }={\mathrm e}^{3 x}+\sin \left (x \right )} \]

1.1.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { {\mathrm e}^{3 x}+\sin \left (x \right )\,\mathop {\mathrm {d}x}}\\ &= \frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \\ \end{align*}

Figure 1: Slope field plot

Verification of solutions

\[ y = \frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \] Verified OK.

1.1.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{3 x}+\sin \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left ({\mathrm e}^{3 x}+\sin \left (x \right )\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \end {array} \]

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: 16

dsolve(diff(y(x),x)=exp(3*x)+sin(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 21

DSolve[y'[x]==Exp[3*x]+Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{3 x}}{3}-\cos (x)+c_1 \]