Internal problem ID [6774]
Internal file name [OUTPUT/6021_Monday_July_25_2022_01_59_46_AM_52332013/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: 8.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {{y^{\prime }}^{2}-x^{2} y^{2}=0} \] The ode \begin {align*} {y^{\prime }}^{2}-x^{2} y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (y x -y^{\prime }\right ) \left (y x +y^{\prime }\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} y x -y^{\prime } = 0\tag {1} \\ y x +y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= x y \end {align*}
Where \(f(x)=x\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= x \,d x\\ \int { \frac {1}{y} \,dy} &= \int {x \,d x}\\ \ln \left (y \right )&=\frac {x^{2}}{2}+c_{1}\\ y&={\mathrm e}^{\frac {x^{2}}{2}+c_{1}}\\ &=c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -x y \end {align*}
Where \(f(x)=-x\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -x \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-x \,d x}\\ \ln \left (y \right )&=-\frac {x^{2}}{2}+c_{2}\\ y&={\mathrm e}^{-\frac {x^{2}}{2}+c_{2}}\\ &=c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{2}-x^{2} y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=y x , y^{\prime }=-y x \right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=y x \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-y x \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-x \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -x d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\frac {x^{2}}{2}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\frac {x^{2}}{2}+c_{1}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y={\mathrm e}^{-\frac {x^{2}}{2}+c_{1}}, y={\mathrm e}^{\frac {x^{2}}{2}+c_{1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear 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: 23
dsolve(diff(y(x),x)^2-x^2*y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= {\mathrm e}^{-\frac {x^{2}}{2}} c_{1} \\ y \left (x \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 38
DSolve[(y'[x])^2-x^2*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-\frac {x^2}{2}} \\ y(x)\to c_1 e^{\frac {x^2}{2}} \\ y(x)\to 0 \\ \end{align*}