Internal problem ID [12595]
Internal file name [OUTPUT/11248_Thursday_October_19_2023_04_43_44_PM_72759423/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 1. Introduction. Exercises page 14
Problem number: 37.
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]
\[ \boxed {{y^{\prime }}^{2}=x^{6}} \] The ode \begin {align*} {y^{\prime }}^{2} = x^{6} \end {align*}
is factored to \begin {align*} \left (-x^{3}+y^{\prime }\right ) \left (x^{3}+y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} -x^{3}+y^{\prime } = 0\tag {1} \\ x^{3}+y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { x^{3}\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{4}}{4}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{4}}{4}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{4}}{4}+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{4}}{4}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{4}}{4}+c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { -x^{3}\,\mathop {\mathrm {d}x}}\\ &= -\frac {x^{4}}{4}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{4}}{4}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{4}}{4}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{4}}{4}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{4}}{4}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}=x^{6} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=x^{3}, y^{\prime }=-x^{3}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=x^{3} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int x^{3}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{4}}{4}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x^{4}}{4}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-x^{3} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -x^{3}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{4}}{4}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{4}}{4}+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\frac {x^{4}}{4}+c_{1} , y=\frac {x^{4}}{4}+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(diff(y(x),x)^2=x^6,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x^{4}}{4}+c_{1} \\ y \left (x \right ) &= -\frac {x^{4}}{4}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 29
DSolve[(y'[x])^2==x^6,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^4}{4}+c_1 \\ y(x)\to \frac {x^4}{4}+c_1 \\ \end{align*}