Internal problem ID [4270]
Internal file name [OUTPUT/3763_Sunday_June_05_2022_10_48_19_AM_15230443/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1049.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime }=0} \] The ode \begin {align*} {y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime } = 0 \end {align*}
is factored to \begin {align*} y^{\prime } \left (y^{2}-y^{\prime }\right ) \left (y^{\prime }+2 x \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } = 0\tag {1} \\ y^{2}-y^{\prime } = 0\tag {2} \\ y^{\prime }+2 x = 0\tag {3} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { 0\,\mathop {\mathrm {d}x}}\\ &= c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}}d y &= x +c_{2}\\ -\frac {1}{y}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {1}{x +c_{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{x +c_{2}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{x +c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{x +c_{2}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{x +c_{2}} \] Verified OK.
Solving ODE (3) Integrating both sides gives \begin {align*} y &= \int { -2 x\,\mathop {\mathrm {d}x}}\\ &= -x^{2}+c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -x^{2}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = -x^{2}+c_{3} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -x^{2}+c_{3} \\ \end{align*}
Verification of solutions
\[ y = -x^{2}+c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=0, y^{\prime }=-2 x , y^{\prime }=y^{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=0 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 0d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-2 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -2 x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x^{2}+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=y^{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{x +c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=c_{1} , y=-\frac {1}{x +c_{1}}, y=-x^{2}+c_{1} \right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature 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: 25
dsolve(diff(y(x),x)^3+(2*x-y(x)^2)*diff(y(x),x)^2-2*x*y(x)^2*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{c_{1} -x} \\ y \left (x \right ) &= -x^{2}+c_{1} \\ y \left (x \right ) &= c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.056 (sec). Leaf size: 31
DSolve[(y'[x])^3 +(2*x-y[x]^2)*(y'[x])^2 -2*x*y[x]^2 y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{x+c_1} \\ y(x)\to c_1 \\ y(x)\to -x^2+c_1 \\ \end{align*}