1.45 problem 45

Internal problem ID [7737]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 45
Date solved : Monday, October 21, 2024 at 04:01:19 PM
CAS classification : [_dAlembert]

Solve

\begin{align*} y^{\prime } y&=1-x {y^{\prime }}^{3} \end{align*}

1.45.1 Solved as first order dAlembert ode

Time used: 0.154 (sec)

Let \(p=y^{\prime }\) the ode becomes

\begin{align*} p y = -x \,p^{3}+1 \end{align*}

Solving for \(y\) from the above results in

\begin{align*} y &= -p^{2} x +\frac {1}{p}\tag {1A} \end{align*}

This has the form

\begin{align*} y=xf(p)+g(p)\tag {*} \end{align*}

Where \(f,g\) are functions of \(p=y'(x)\). The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. \(x\) gives

\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}

Comparing the form \(y=x f + g\) to (1A) shows that

\begin{align*} f &= -p^{2}\\ g &= \frac {1}{p} \end{align*}

Hence (2) becomes

\begin{align*} p^{2}+p = \left (-2 x p -\frac {1}{p^{2}}\right ) p^{\prime }\left (x \right )\tag {2A} \end{align*}

The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives

\begin{align*} p^{2}+p = 0 \end{align*}

Solving the above for \(p\) results in

\begin{align*} p_{1} &=-1\\ p_{2} &=0 \end{align*}

Substituting these in (1A) and keeping singular solution that verifies the ode gives

\begin{align*} y = -x -1 \end{align*}

The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in

\begin{align*} p^{\prime }\left (x \right ) = \frac {p \left (x \right )^{2}+p \left (x \right )}{-2 p \left (x \right ) x -\frac {1}{p \left (x \right )^{2}}}\tag {3} \end{align*}

Inverting the above ode gives

\begin{align*} \frac {d}{d p}x \left (p \right ) = \frac {-2 x \left (p \right ) p -\frac {1}{p^{2}}}{p^{2}+p}\tag {4} \end{align*}

This ODE is now solved for \(x \left (p \right )\). The integrating factor is

\begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p +1}d p}\\ \mu &= \left (p +1\right )^{2}\\ \mu &= \left (p +1\right )^{2}\tag {5} \end{align*}

Integrating gives

\begin{align*} x \left (p \right )&= \frac {1}{\mu } \left ( \int { \mu \left (-\frac {1}{p^{3} \left (p +1\right )}\right ) \,dp} + c_1\right )\\ &= \frac {1}{\mu } \left (\frac {\frac {1}{2 p^{2}}+\frac {1}{p}+c_1}{\left (p +1\right )^{2}}+c_1\right ) \\ &= \frac {\frac {1}{2 p^{2}}+\frac {1}{p}+c_1}{\left (p +1\right )^{2}}\tag {5} \end{align*}

Now we need to eliminate \(p\) between the above solution and (1A). The first method is to solve for \(p\) from Eq. (1A) and substitute the result into Eq. (5). The Second method is to solve for \(p\) from Eq. (5) and substitute the result into (1A).

Eliminating \(p\) from the following two equations

\begin{align*} x &= \frac {\frac {1}{2 p^{2}}+\frac {1}{p}+c_1}{\left (p +1\right )^{2}} \\ y &= -p^{2} x +\frac {1}{p} \\ \end{align*}

results in

\begin{align*} p &= \operatorname {RootOf}\left (-1+2 x \,\textit {\_Z}^{4}+4 x \,\textit {\_Z}^{3}+\left (-2 c_1 +2 x \right ) \textit {\_Z}^{2}-2 \textit {\_Z} \right ) \\ \end{align*}

Substituting the above into Eq (1A) and simplifying gives

\begin{align*} y &= -\frac {x \operatorname {RootOf}\left (-1+2 x \,\textit {\_Z}^{4}+4 x \,\textit {\_Z}^{3}+\left (-2 c_1 +2 x \right ) \textit {\_Z}^{2}-2 \textit {\_Z} \right )^{3}-1}{\operatorname {RootOf}\left (-1+2 x \,\textit {\_Z}^{4}+4 x \,\textit {\_Z}^{3}+\left (-2 c_1 +2 x \right ) \textit {\_Z}^{2}-2 \textit {\_Z} \right )} \\ \end{align*}

1.45.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y=1-x {y^{\prime }}^{3} \\ \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 }=\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}-\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}, y^{\prime }=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{12 x}+\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}+\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}\right )}{2}, y^{\prime }=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{12 x}+\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}+\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}-\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{12 x}+\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}+\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{12 x}+\frac {y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}{6 x}+\frac {2 y}{{\left (\left (12 \sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+108\right ) x^{2}\right )}^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]

1.45.3 Maple trace

Methods for first order ODEs:
 

1.45.4 Maple dsolve solution

Solving time : 0.050 (sec)
Leaf size : 1792

dsolve(diff(y(x),x)*y(x) = 1-x*diff(y(x),x)^3, 
       y(x),singsol=all)
 

\begin{align*} \frac {12 c_1 \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} y+\left (\frac {3^{{2}/{3}} 2^{{1}/{3}} \left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}}{6}+2^{{2}/{3}} 3^{{1}/{3}} y^{2}\right ) x \right ) x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}}{\left (2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}-2 x \left (y 2^{{1}/{3}} 3^{{2}/{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )\right )^{2} \left (y 2^{{2}/{3}} 3^{{1}/{3}} x -{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}}+x +\frac {9 \,3^{{1}/{3}} 2^{{2}/{3}} x^{4} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}} \left (3^{{2}/{3}} 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}} y-\frac {2^{{2}/{3}} 3^{{5}/{6}} \sqrt {\frac {4 y^{3}+27 x}{x}}\, x}{2}-\frac {9 \,3^{{1}/{3}} 2^{{2}/{3}} x}{2}-\frac {3 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}}{2}\right )}{\left (-\frac {2^{{2}/{3}} 3^{{1}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{2}+x \left (y 2^{{1}/{3}} 3^{{2}/{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}\right )\right )^{2} \left (-y 2^{{2}/{3}} 3^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}} &= 0 \\ -\frac {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} c_1 \left (-8 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} y+\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-2 y^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) x \right ) x^{3}}{6 {\left (\left (-i+\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{1}/{3}}+3^{{5}/{6}}\right ) y x \right )}^{2} \left (\frac {2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{6}+\left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) y 2^{{1}/{3}}\right ) x \right )^{2}}+x +\frac {24 {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}} 3^{{1}/{3}} 2^{{2}/{3}} x^{4} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\frac {x \left (i 3^{{1}/{3}}+\frac {3^{{5}/{6}}}{3}\right ) 2^{{2}/{3}} \sqrt {\frac {4 y^{3}+27 x}{x}}}{2}+\frac {3 x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}}{2}\right )}{{\left (\left (-i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+\left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) y 2^{{2}/{3}} x \right )}^{2} \left (\frac {2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{6}+\left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) y 2^{{1}/{3}}\right ) x \right )^{2}} &= 0 \\ \frac {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} c_1 \left (8 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}} y+\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}-2 y^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) x \right ) x^{3}}{6 {\left (\frac {\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+y \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}}\right ) x \right )}^{2} {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) y x \right )}^{2}}+x +\frac {24 {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}} 3^{{1}/{3}} 2^{{2}/{3}} x^{4} \left ({\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+y \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+\frac {x \left (i 3^{{1}/{3}}-\frac {3^{{5}/{6}}}{3}\right ) 2^{{2}/{3}} \sqrt {\frac {4 y^{3}+27 x}{x}}}{2}+\frac {3 x \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}}}{2}\right )}{{\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{2}/{3}}+\left (-i 3^{{1}/{3}}+3^{{5}/{6}}\right ) y 2^{{2}/{3}} x \right )}^{2} {\left (\frac {\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{{1}/{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{{1}/{3}}+y \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) 2^{{1}/{3}}\right ) x \right )}^{2}} &= 0 \\ \end{align*}

1.45.5 Mathematica DSolve solution

Solving time : 85.048 (sec)
Leaf size : 20717

DSolve[{D[y[x],x]*y[x]==1-x*(D[y[x],x])^3,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

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