Internal problem ID [4406]
Internal file name [OUTPUT/3899_Sunday_June_05_2022_11_37_32_AM_31421343/index.tex]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 7
Problem number: 1.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{2}-5 y^{\prime }=-6} \] The ode \begin {align*} {y^{\prime }}^{2}-5 y^{\prime } = -6 \end {align*}
is factored to \begin {align*} \left (y^{\prime }-2\right ) \left (y^{\prime }-3\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-2 = 0\tag {1} \\ y^{\prime }-3 = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { 2\,\mathop {\mathrm {d}x}}\\ &= 2 x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = 2 x +c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = 2 x +c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { 3\,\mathop {\mathrm {d}x}}\\ &= 3 x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 x +c_{2} \\ \end{align*}
Verification of solutions
\[ y = 3 x +c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 x +c_{2} \\ \end{align*}
Verification of solutions
\[ y = 3 x +c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-5 y^{\prime }=-6 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=2, y^{\prime }=3\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 2d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=2 x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=3 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 3d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=3 x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=3 x +c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=2 x +c_{1} , y=3 x +c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve((diff(y(x),x))^2-5*diff(y(x),x)+6=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 3 x +c_{1} \\ y \left (x \right ) &= 2 x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 21
DSolve[(y'[x])^2-5*y'[x]+6==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 x+c_1 \\ y(x)\to 3 x+c_1 \\ \end{align*}