Internal problem ID [4141]
Internal file name [OUTPUT/3634_Sunday_June_05_2022_09_52_32_AM_34699434/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 905.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "exact", "linear", "separable", "differentialType", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"
Maple gives the following as the ode type
[_separable]
\[ \boxed {x^{2} {y^{\prime }}^{2}+3 x y^{\prime } y+2 y^{2}=0} \] The ode \begin {align*} x^{2} {y^{\prime }}^{2}+3 x y^{\prime } y+2 y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (x y^{\prime }+y\right ) \left (x y^{\prime }+2 y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} x y^{\prime }+y = 0\tag {1} \\ x y^{\prime }+2 y = 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)\\ &= -\frac {y}{x} \end {align*}
Where \(f(x)=-\frac {1}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {1}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {1}{x} \,d x}\\ \ln \left (y \right )&=-\ln \left (x \right )+c_{1}\\ y&={\mathrm e}^{-\ln \left (x \right )+c_{1}}\\ &=\frac {c_{1}}{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{x} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{x} \] Verified OK.
Solving ODE (2) In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {2 y}{x} \end {align*}
Where \(f(x)=-\frac {2}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {2}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {2}{x} \,d x}\\ \ln \left (y \right )&=-2 \ln \left (x \right )+c_{2}\\ y&={\mathrm e}^{-2 \ln \left (x \right )+c_{2}}\\ &=\frac {c_{2}}{x^{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{2}}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{2}}{x^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} {y^{\prime }}^{2}+3 x y^{\prime } y+2 y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {2 y}{x}, y^{\prime }=-\frac {y}{x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {2 y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {2}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {2}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-2 \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{x^{2}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{x} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {{\mathrm e}^{c_{1}}}{x^{2}}, y=\frac {{\mathrm e}^{c_{1}}}{x}\right \} \end {array} \]
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: 17
dsolve(x^2*diff(y(x),x)^2+3*x*diff(y(x),x)*y(x)+2*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {c_{1}}{x} \\ y \left (x \right ) &= \frac {c_{1}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 26
DSolve[x^2 (y'[x])^2+3 x y[x] y'[x]+2 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1}{x^2} \\ y(x)\to \frac {c_1}{x} \\ y(x)\to 0 \\ \end{align*}