Internal problem ID [8805]
Internal file name [OUTPUT/7740_Sunday_June_05_2022_11_52_35_PM_93563058/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 471.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "quadrature", "separable", "differentialType", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }=x} \] The ode \begin {align*} y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime } = x \end {align*}
is factored to \begin {align*} \left (y^{\prime }-1\right ) \left (y^{\prime } y+x \right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime }-1 = 0\tag {1} \\ y^{\prime } y+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 { 1\,\mathop {\mathrm {d}x}}\\ &= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = x +c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = x +c_{1} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x}{y} \end {align*}
Where \(f(x)=-x\) and \(g(y)=\frac {1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{y}} \,dy &= -x \,d x \\ \int { \frac {1}{\frac {1}{y}} \,dy} &= \int {-x \,d x} \\ \frac {y^{2}}{2}&=-\frac {x^{2}}{2}+c_{2} \\ \end{align*} Which results in \begin{align*} y &= \sqrt {-x^{2}+2 c_{2}} \\ y &= -\sqrt {-x^{2}+2 c_{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-x^{2}+2 c_{2}} \\ \tag{2} y &= -\sqrt {-x^{2}+2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-x^{2}+2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-x^{2}+2 c_{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-x^{2}+2 c_{2}} \\ \tag{2} y &= -\sqrt {-x^{2}+2 c_{2}} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-x^{2}+2 c_{2}} \] Verified OK.
\[ y = -\sqrt {-x^{2}+2 c_{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y {y^{\prime }}^{2}-\left (y-x \right ) y^{\prime }=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=1, y^{\prime }=-\frac {x}{y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {x}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=-x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int -x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-x^{2}+2 c_{1}}, y=-\sqrt {-x^{2}+2 c_{1}}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=x +c_{1} , \left \{y=\sqrt {-x^{2}+2 c_{1}}, y=-\sqrt {-x^{2}+2 c_{1}}\right \}\right \} \end {array} \]
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 trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(y(x)*diff(y(x),x)^2-(y(x)-x)*diff(y(x),x)-x = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.117 (sec). Leaf size: 47
DSolve[-x - (-x + y[x])*y'[x] + y[x]*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+c_1 \\ y(x)\to -\sqrt {-x^2+2 c_1} \\ y(x)\to \sqrt {-x^2+2 c_1} \\ \end{align*}