Internal problem ID [4166]
Internal file name [OUTPUT/3659_Sunday_June_05_2022_10_05_40_AM_25685542/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 932.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {4 x \left (a -x \right ) \left (b -x \right ) {y^{\prime }}^{2}=\left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2}} \] Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \tag {1} \\ y^{\prime }&=-\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} y &= \int { \frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\,\mathop {\mathrm {d}x}}\\ &= \frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}-\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (-\frac {a}{3}-\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}-\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (-\frac {a}{3}-\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}-\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (-\frac {a}{3}-\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} y &= \int { -\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}+\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\frac {a}{3}+\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}+\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\frac {a}{3}+\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}+\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \left (\frac {a}{3}+\frac {b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \operatorname {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \operatorname {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 x \left (a -x \right ) \left (b -x \right ) {y^{\prime }}^{2}=\left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2} \\ \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 {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}, y^{\prime }=\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\left (-2 a -2 b \right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}+\frac {2 \left (\frac {2 a}{3}+\frac {2 b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a^{3}+2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}-a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}-2 a b x +2 a \,x^{2}+2 b \,x^{2}-2 x^{3}+3 c_{1} \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {a b -2 a x -2 b x +2 x^{2}}{2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {\left (-2 a -2 b \right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}-\frac {a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {2 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3}-\frac {2 \left (\frac {2 a}{3}+\frac {2 b}{3}\right ) a \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \left (\left (a -b \right ) \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+b \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )\right )}{\sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a^{3}-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}+a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}+2 a b x -2 a \,x^{2}-2 b \,x^{2}+2 x^{3}+3 c_{1} \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\frac {-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a^{3}+2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}-a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}-2 a b x +2 a \,x^{2}+2 b \,x^{2}-2 x^{3}+3 c_{1} \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}, y=\frac {2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a^{3}-2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticE}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}+a^{2} b \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \sqrt {\frac {x}{a}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right )+2 \sqrt {-\frac {-a +x}{a}}\, \sqrt {\frac {-b +x}{a -b}}\, \mathit {EllipticF}\left (\sqrt {-\frac {-a +x}{a}}, \sqrt {\frac {a}{a -b}}\right ) \sqrt {\frac {x}{a}}\, a \,b^{2}+2 a b x -2 a \,x^{2}-2 b \,x^{2}+2 x^{3}+3 c_{1} \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}{3 \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables <- differential order: 1; missing y(x) successful`
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 85
dsolve(4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (\int \frac {2 x^{2}+\left (-2 a -2 b \right ) x +a b}{\sqrt {x \left (-x +b \right ) \left (a -x \right )}}d x \right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {2 x^{2}+\left (-2 a -2 b \right ) x +a b}{\sqrt {x \left (-x +b \right ) \left (a -x \right )}}d x \right )}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.208 (sec). Leaf size: 375
DSolve[4 x(a-x)(b-x) (y'[x])^2==(a b-2 x(a+b)+2 x^2)^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1-\frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}} \\ y(x)\to \frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}}+c_1 \\ \end{align*}