Internal problem ID [13352]
Internal file name [OUTPUT/12524_Wednesday_February_14_2024_11_54_40_PM_8658397/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 5. LINEAR FIRST ORDER EQUATIONS. Additional exercises. page
103
Problem number: 5.1 (g).
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]
\[ \boxed {y^{\prime }={\mathrm e}^{2 x}} \]
Integrating both sides gives \begin {align*} y &= \int { {\mathrm e}^{2 x}\,\mathop {\mathrm {d}x}}\\ &= \frac {{\mathrm e}^{2 x}}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{2 x}}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{2 x}}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{2 x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int {\mathrm e}^{2 x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {{\mathrm e}^{2 x}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{2 x}}{2}+c_{1} \end {array} \]
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: 12
dsolve(diff(y(x),x)-exp(2*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x}}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 17
DSolve[y'[x]-Exp[2*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{2 x}}{2}+c_1 \]