Internal problem ID [6156]
Internal file name [OUTPUT/5404_Sunday_June_05_2022_03_36_24_PM_12653789/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 4.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_integrable_as_is", "second_order_ode_missing_y", "exact nonlinear second order ode"
Maple gives the following as the ode type
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]
\[ \boxed {y^{\prime \prime } y^{\prime }=x \left (1+x \right )} \]
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int y^{\prime \prime } y^{\prime }d x &= \int \left (x^{2}+x \right )d x\\ \frac {{y^{\prime }}^{2}}{2} = \frac {1}{3} x^{3}+\frac {1}{2} x^{2} + c_{1} \end {align*}
Which is now solved for \(y\). Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \\ \tag{2} y &= \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \] Verified OK.
\[ y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \] Verified OK.
This is second order ode with missing dependent variable \(y\). Let \begin {align*} p(x) &= y^{\prime } \end {align*}
Then \begin {align*} p'(x) &= y^{\prime \prime } \end {align*}
Hence the ode becomes \begin {align*} p \left (x \right ) p^{\prime }\left (x \right )-x^{2}-x = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= \frac {x \left (1+x \right )}{p} \end {align*}
Where \(f(x)=x \left (1+x \right )\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= x \left (1+x \right ) \,d x \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {x \left (1+x \right ) \,d x} \\ \frac {p^{2}}{2}&=\frac {1}{3} x^{3}+\frac {1}{2} x^{2}+c_{1} \\ \end{align*} The solution is \[ \frac {p \left (x \right )^{2}}{2}-\frac {x^{3}}{3}-\frac {x^{2}}{2}-c_{1} = 0 \] For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {{y^{\prime }}^{2}}{2}-\frac {x^{3}}{3}-\frac {x^{2}}{2}-c_{1} = 0 \end {align*}
Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \\ \tag{2} y &= \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \] Verified OK.
\[ y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \] Verified OK.
An exact non-linear second order ode has the form \begin {align*} a_{2} \left (x , y, y^{\prime }\right ) y^{\prime \prime }+a_{1} \left (x , y, y^{\prime }\right ) y^{\prime }+a_{0} \left (x , y, y^{\prime }\right )&=0 \end {align*}
Where the following conditions are satisfied \begin {align*} \frac {\partial a_2}{\partial y} &= \frac {\partial a_1}{\partial y'}\\ \frac {\partial a_2}{\partial x} &= \frac {\partial a_0}{\partial y'}\\ \frac {\partial a_1}{\partial x} &= \frac {\partial a_0}{\partial y} \end {align*}
Looking at the the ode given we see that \begin {align*} a_2 &= y^{\prime }\\ a_1 &= 0\\ a_0 &= -x^{2}-x \end {align*}
Applying the conditions to the above shows this is a nonlinear exact second order ode. Therefore it can be reduced to first order ode given by \begin {align*} \int {a_2\,d y'} + \int {a_1\,d y} + \int {a_0\,d x} &= c_{1}\\ \int {y^{\prime }\,d y'} + \int {0\,d y} + \int {-x^{2}-x\,d x} &= c_{1} \end {align*}
Which results in \begin {align*} \frac {{y^{\prime }}^{2}}{2}-\frac {x^{2} \left (2 x +3\right )}{6} = c_{1} \end {align*}
Which is now solved Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {1} \\ y^{\prime }&=-\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \\ \tag{2} y &= \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \] Verified OK.
\[ y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, diff(_b(_a), _a) = _a*(_a+1)/_b(_a), _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 51
dsolve(diff(y(x),x$2)*diff(y(x),x)=x*(1+x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (\int \sqrt {6 x^{3}+9 x^{2}+9 c_{1}}d x \right )}{3}+c_{2} \\ y \left (x \right ) &= \frac {\left (\int \sqrt {6 x^{3}+9 x^{2}+9 c_{1}}d x \right )}{3}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 61.466 (sec). Leaf size: 12885
DSolve[y''[x]*y'[x]==x*(1+x),y[x],x,IncludeSingularSolutions -> True]
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