Internal problem ID [12660]
Internal file name [OUTPUT/11313_Friday_November_03_2023_06_30_19_AM_14899875/index.tex
]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.3.1, page 57
Problem number: 1.
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]
\[ \boxed {y^{\prime }=3 x +1} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}
Where here \begin {align*} p(x) &=0\\ q(x) &=3 x +1 \end {align*}
Hence the ode is \begin {align*} y^{\prime } = 3 x +1 \end {align*}
The domain of \(p(x)=0\) is \[
\{-\infty
Integrating both sides gives \begin {align*} y &= \int { 3 x +1\,\mathop {\mathrm {d}x}}\\ &= \frac {3}{2} x^{2}+x +c_{1} \end {align*}
Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=2\) in the above solution gives an
equation to solve for the constant of integration. \begin {align*} 2 = \frac {5}{2}+c_{1} \end {align*}
The solutions are \begin {align*} c_{1} = -{\frac {1}{2}} \end {align*}
Trying the constant \begin {align*} c_{1} = -{\frac {1}{2}} \end {align*}
Substituting this in the general solution gives \begin {align*} y&=\frac {3}{2} x^{2}+x -\frac {1}{2} \end {align*}
The constant \(c_{1} = -{\frac {1}{2}}\) gives valid solution.
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= \frac {3}{2} x^{2}+x -\frac {1}{2} \\
\end{align*} Verification of solutions
\[
y = \frac {3}{2} x^{2}+x -\frac {1}{2}
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=3 x +1, y \left (1\right )=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (3 x +1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {3}{2} x^{2}+x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {3}{2} x^{2}+x +c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=2 \\ {} & {} & 2=\frac {5}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =-\frac {1}{2} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =-\frac {1}{2}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {3}{2} x^{2}+x -\frac {1}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {3}{2} x^{2}+x -\frac {1}{2} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
\[
y \left (x \right ) = \frac {3}{2} x^{2}+x -\frac {1}{2}
\]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 17
\[
y(x)\to \frac {3 x^2}{2}+x-\frac {1}{2}
\]
5.1.2 Solving as quadrature ode
5.1.3 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
<- quadrature successful`
dsolve([diff(y(x),x)=3*x+1,y(1) = 2],y(x), singsol=all)
DSolve[{y'[x]==3*x+1,{y[1]==2}},y[x],x,IncludeSingularSolutions -> True]