Internal problem ID [4042]
Internal file name [OUTPUT/3535_Sunday_June_05_2022_09_35_58_AM_32424933/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 28
Problem number: 802.
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]
\[ \boxed {{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime }=0} \] The ode \begin {align*} {y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime } = 0 \end {align*}
is factored to \begin {align*} y^{\prime } \left (-2 x^{2}+y^{\prime }+2 x \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } = 0\tag {1} \\ -2 x^{2}+y^{\prime }+2 x = 0\tag {2} \\ \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*} y &= \int { 2 x^{2}-2 x\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{2} \left (2 x -3\right )}{3}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=0, y^{\prime }=2 x^{2}-2 x \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=0 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 0d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=2 x^{2}-2 x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (2 x^{2}-2 x \right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {2}{3} x^{3}-x^{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2}{3} x^{3}-x^{2}+c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=c_{1} , y=\frac {2}{3} x^{3}-x^{2}+c_{1} \right \} \end {array} \]
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 trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(diff(y(x),x)^2-2*x^2*diff(y(x),x)+2*x*diff(y(x),x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2}{3} x^{3}-x^{2}+c_{1} \\ y \left (x \right ) &= c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 26
DSolve[(y'[x])^2-2 x^2 y'[x]+2 x y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \\ y(x)\to \frac {2 x^3}{3}-x^2+c_1 \\ \end{align*}