Internal problem ID [5087]
Internal file name [OUTPUT/4580_Sunday_June_05_2022_03_01_16_PM_65589201/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x \left (y-3\right ) y^{\prime }-4 y=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {4 y}{x \left (y -3\right )} \end {align*}
Where \(f(x)=\frac {4}{x}\) and \(g(y)=\frac {y}{y -3}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y}{y -3}} \,dy &= \frac {4}{x} \,d x \\ \int { \frac {1}{\frac {y}{y -3}} \,dy} &= \int {\frac {4}{x} \,d x} \\ y -3 \ln \left (y \right )&=4 \ln \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= {\mathrm e}^{-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}}}{3}\right )-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}}}{3}\right )-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}}}{3}\right )-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (y-3\right ) y^{\prime }-4 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {4 y}{x \left (y-3\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (y-3\right )}{y}=\frac {4}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (y-3\right )}{y}d x =\int \frac {4}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y-3 \ln \left (y\right )=4 \ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\mathit {LambertW}\left (-\frac {{\mathrm e}^{-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}}}{3}\right )-\frac {4 \ln \left (x \right )}{3}-\frac {c_{1}}{3}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 16
dsolve(x*(y(x)-3)*diff(y(x),x)=4*y(x),y(x), singsol=all)
\[ y \left (x \right ) = -3 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {4 c_{1}}{3}}}{3 x^{\frac {4}{3}}}\right ) \]
✓ Solution by Mathematica
Time used: 13.068 (sec). Leaf size: 94
DSolve[x*(y[x]-3)*y'[x]==4*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -3 W\left (\frac {1}{3} \sqrt [3]{-\frac {e^{-c_1}}{x^4}}\right ) \\ y(x)\to -3 W\left (-\frac {1}{3} \sqrt [3]{-1} \sqrt [3]{-\frac {e^{-c_1}}{x^4}}\right ) \\ y(x)\to -3 W\left (\frac {1}{3} (-1)^{2/3} \sqrt [3]{-\frac {e^{-c_1}}{x^4}}\right ) \\ y(x)\to 0 \\ \end{align*}