Internal problem ID [6785]
Internal file name [OUTPUT/6032_Tuesday_July_26_2022_05_04_44_AM_54275792/index.tex
]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES
Page 309
Problem number: 19.
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]
\[ \boxed {y {y^{\prime }}^{2} x +\left (x +y\right ) y^{\prime }=-1} \] The ode \begin {align*} y {y^{\prime }}^{2} x +\left (x +y\right ) y^{\prime } = -1 \end {align*}
is factored to \begin {align*} \left (y^{\prime } x +1\right ) \left (y^{\prime } y+1\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y^{\prime } x +1 = 0\tag {1} \\ y^{\prime } y+1 = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Integrating both sides gives \begin {align*} y &= \int { -\frac {1}{x}\,\mathop {\mathrm {d}x}}\\ &= -\ln \left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\ln \left (x \right )+c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\ln \left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\ln \left (x \right )+c_{1} \] Verified OK.
Solving ODE (2) Integrating both sides gives \begin {align*} \int -y d y &= x +c_{2}\\ -\frac {y^{2}}{2}&=x +c_{2} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {-2 c_{2} -2 x}\\ y_2&=-\sqrt {-2 c_{2} -2 x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 c_{2} -2 x} \\ \tag{2} y &= -\sqrt {-2 c_{2} -2 x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 c_{2} -2 x} \] Verified OK.
\[ y = -\sqrt {-2 c_{2} -2 x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {-2 c_{2} -2 x} \\ \tag{2} y &= -\sqrt {-2 c_{2} -2 x} \\ \end{align*}
Verification of solutions
\[ y = \sqrt {-2 c_{2} -2 x} \] Verified OK.
\[ y = -\sqrt {-2 c_{2} -2 x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y {y^{\prime }}^{2} x +\left (x +y\right ) y^{\prime }=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {1}{x}, y^{\prime }=-\frac {1}{y}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (x \right )+c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{y} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } yd x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}=-x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {2 c_{1} -2 x}, y=-\sqrt {2 c_{1} -2 x}\right \} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\ln \left (x \right )+c_{1} , \left \{y=\sqrt {2 c_{1} -2 x}, y=-\sqrt {2 c_{1} -2 x}\right \}\right \} \end {array} \]
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 trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 32
dsolve(x*y(x)*diff(y(x),x)^2+(x+y(x))*diff(y(x),x)+1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \sqrt {c_{1} -2 x} \\ y \left (x \right ) &= -\sqrt {c_{1} -2 x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 53
DSolve[x*y[x]*(y'[x])^2+(x+y[x])*y'[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {2} \sqrt {-x+c_1} \\ y(x)\to \sqrt {2} \sqrt {-x+c_1} \\ y(x)\to -\log (x)+c_1 \\ \end{align*}