15.18 problem 426

Internal problem ID [3680]
Internal file name [OUTPUT/3173_Sunday_June_05_2022_08_54_42_AM_28406630/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 15
Problem number: 426.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y y^{\prime }-\operatorname {a1} y-\operatorname {a2} y^{2}=\operatorname {a0}} \]

15.18.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {y}{\operatorname {a2} \,y^{2}+\operatorname {a1} y +\operatorname {a0}}d y &= \int {dx}\\ \int _{}^{y}\frac {\textit {\_a}}{\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a}&= x +c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {\textit {\_a}}{\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} &= x +c_{1} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {\textit {\_a}}{\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} = x +c_{1} \] Verified OK.

15.18.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\mathit {a1} y-\mathit {a2} y^{2}=\mathit {a0} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\mathit {a0} +\mathit {a1} y+\mathit {a2} y^{2}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\mathit {a0} +\mathit {a1} y+\mathit {a2} y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\mathit {a0} +\mathit {a1} y+\mathit {a2} y^{2}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (\mathit {a0} +\mathit {a1} y+\mathit {a2} y^{2}\right )}{2 \mathit {a2}}-\frac {\mathit {a1} \arctan \left (\frac {2 y \mathit {a2} +\mathit {a1}}{\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}\right )}{\mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}=x +c_{1} \\ \bullet & {} & \textrm {Convert}\hspace {3pt} \arctan \mathrm {to \esapos ln\esapos } \\ {} & {} & \frac {\ln \left (\mathit {a0} +\mathit {a1} y+\mathit {a2} y^{2}\right )}{2 \mathit {a2}}-\frac {\mathrm {I} \mathit {a1} \left (\ln \left (1-\frac {\mathrm {I} \left (2 y \mathit {a2} +\mathit {a1} \right )}{\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}\right )-\ln \left (1+\frac {\mathrm {I} \left (2 y \mathit {a2} +\mathit {a1} \right )}{\sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}\right )\right )}{2 \mathit {a2} \sqrt {4 \mathit {a0} \mathit {a2} -\mathit {a1}^{2}}}=x +c_{1} \end {array} \]

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: 0.563 (sec). Leaf size: 222

dsolve(y(x)*diff(y(x),x) = a0+a1*y(x)+a2*y(x)^2,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {4 \tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a0} \operatorname {a2} -\tan \left (\operatorname {RootOf}\left (2 c_{1} \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 x \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}+2 \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (2\right )-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}\right )}{\operatorname {a2}}\right )+2 \textit {\_Z} \operatorname {a1} \right )\right ) \operatorname {a1}^{2}-\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}\, \operatorname {a1}}{2 \operatorname {a2} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}} \]

✓ Solution by Mathematica

Time used: 0.385 (sec). Leaf size: 123

DSolve[y[x] y'[x]==a0+a1 y[x]+a2 y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\log (\text {$\#$1} (\text {$\#$1} \text {a2}+\text {a1})+\text {a0})-\frac {2 \text {a1} \arctan \left (\frac {2 \text {$\#$1} \text {a2}+\text {a1}}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}\right )}{\sqrt {4 \text {a0} \text {a2}-\text {a1}^2}}}{2 \text {a2}}\&\right ][x+c_1] \\ y(x)\to \frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}}{2 \text {a2}} \\ y(x)\to -\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}}{2 \text {a2}} \\ \end{align*}