Internal problem ID [4143]
Internal file name [OUTPUT/3636_Sunday_June_05_2022_09_52_49_AM_51234384/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 907.
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]
\[ \boxed {x^{2} {y^{\prime }}^{2}+4 x y^{\prime } y-5 y^{2}=0} \] The ode \begin {align*} x^{2} {y^{\prime }}^{2}+4 x y^{\prime } y-5 y^{2} = 0 \end {align*}
is factored to \begin {align*} \left (x y^{\prime }+5 y\right ) \left (-x y^{\prime }+y\right ) = 0 \end {align*}
Which gives the following equations \begin {align*} x y^{\prime }+5 y = 0\tag {1} \\ -x y^{\prime }+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 {5 y}{x} \end {align*}
Where \(f(x)=-\frac {5}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {5}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {5}{x} \,d x}\\ \ln \left (y \right )&=-5 \ln \left (x \right )+c_{1}\\ y&={\mathrm e}^{-5 \ln \left (x \right )+c_{1}}\\ &=\frac {c_{1}}{x^{5}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{x^{5}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{x^{5}} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1}}{x^{5}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1}}{x^{5}} \] Verified OK.
Solving ODE (2) 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_{2}\\ y&={\mathrm e}^{\ln \left (x \right )+c_{2}}\\ &=c_{2} x \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x \\ \end{align*}
Verification of solutions
\[ y = c_{2} x \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x \\ \end{align*}
Verification of solutions
\[ y = c_{2} x \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} {y^{\prime }}^{2}+4 x y^{\prime } y-5 y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {y}{x}, y^{\prime }=-\frac {5 y}{x}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=\frac {1}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int \frac {1}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=\ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=x \,{\mathrm e}^{c_{1}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {5 y}{x} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=-\frac {5}{x} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int -\frac {5}{x}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=-5 \ln \left (x \right )+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{c_{1}}}{x^{5}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=x \,{\mathrm e}^{c_{1}}, y=\frac {{\mathrm e}^{c_{1}}}{x^{5}}\right \} \end {array} \]
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: 0.015 (sec). Leaf size: 15
dsolve(x^2*diff(y(x),x)^2+4*x*diff(y(x),x)*y(x)-5*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= c_{1} x \\ y \left (x \right ) &= \frac {c_{1}}{x^{5}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 24
DSolve[x^2 (y'[x])^2+4 x y[x] y'[x]-5 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1}{x^5} \\ y(x)\to c_1 x \\ y(x)\to 0 \\ \end{align*}