Internal problem ID [6148]
Internal file name [OUTPUT/5396_Sunday_June_05_2022_03_36_12_PM_71582804/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(j).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x y^{2}-y^{\prime } x^{2}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y^{2}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=y^{2}\). Integrating both sides gives \begin{align*} \frac {1}{y^{2}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{y^{2}} \,dy} &= \int {\frac {1}{x} \,d x} \\ -\frac {1}{y}&=\ln \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= -\frac {1}{\ln \left (x \right )+c_{1}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{\ln \left (x \right )+c_{1}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{\ln \left (x \right )+c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{2}-y^{\prime } x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}}{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int \frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{\ln \left (x \right )+c_{1}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 12
dsolve(x*y(x)^2-diff(y(x),x)*x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{-\ln \left (x \right )+c_{1}} \]
✓ Solution by Mathematica
Time used: 0.107 (sec). Leaf size: 19
DSolve[x*y[x]^2-y'[x]*x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{\log (x)+c_1} \\ y(x)\to 0 \\ \end{align*}