3.5 problem 59

Internal problem ID [3323]
Internal file name [OUTPUT/2815_Sunday_June_05_2022_08_41_02_AM_5287664/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 3
Problem number: 59.
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]

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

3.5.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{\operatorname {a2} \,y^{2}+\operatorname {a1} y +\operatorname {a0}}d y &= x +c_{1}\\ \frac {2 \arctan \left (\frac {2 \operatorname {a2} y +\operatorname {a1}}{\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}\right )}{\sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}&=x +c_{1} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\tan \left (\frac {c_{1} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}+\frac {x \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}\right ) \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}-\operatorname {a1}}{2 \operatorname {a2}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\tan \left (\frac {c_{1} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}+\frac {x \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}\right ) \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}-\operatorname {a1}}{2 \operatorname {a2}} \\ \end{align*}

Verification of solutions

\[ y = \frac {\tan \left (\frac {c_{1} \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}+\frac {x \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}}{2}\right ) \sqrt {4 \operatorname {a0} \operatorname {a2} -\operatorname {a1}^{2}}-\operatorname {a1}}{2 \operatorname {a2}} \] Verified OK.

3.5.2 Maple step by step solution

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

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: 0.016 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 32.049 (sec). Leaf size: 106

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

\begin{align*} y(x)\to \frac {-\text {a1}+\sqrt {4 \text {a0} \text {a2}-\text {a1}^2} \tan \left (\frac {1}{2} (x+c_1) \sqrt {4 \text {a0} \text {a2}-\text {a1}^2}\right )}{2 \text {a2}} \\ 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*}