1.548 problem 551

Internal problem ID [8885]
Internal file name [OUTPUT/7820_Monday_June_06_2022_12_36_12_AM_92947488/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 551.
ODE order: 1.
ODE degree: 0.

The type(s) of ODE detected by this program : "separable", "first_order_nonlinear_p_but_separable"

Maple gives the following as the ode type

[_separable]

\[ \boxed {{y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1}=0} \]

1.548.1 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \left (y -a \right )^{\frac {n +1}{n}} \left (y -b \right )^{\frac {n -1}{n}} f \left (x \right ) \end {align*}

Where \(f(x)=f \left (x \right )\) and \(g(y)=\left (y -a \right )^{\frac {n +1}{n}} \left (y -b \right )^{\frac {n -1}{n}}\). Integrating both sides gives \begin{align*} \frac {1}{\left (y -a \right )^{\frac {n +1}{n}} \left (y -b \right )^{\frac {n -1}{n}}} \,dy &= f \left (x \right ) \,d x \\ \int { \frac {1}{\left (y -a \right )^{\frac {n +1}{n}} \left (y -b \right )^{\frac {n -1}{n}}} \,dy} &= \int {f \left (x \right ) \,d x} \\ -\frac {n \left (a -y \right ) \left (b -y \right ) \left (y -a \right )^{-\frac {n +1}{n}} \left (y -b \right )^{-\frac {n -1}{n}}}{a -b}&=\int f \left (x \right )d x +c_{1} \\ \end{align*} The solution is \[ -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (y-a \right )^{-\frac {n +1}{n}} \left (y-b \right )^{-\frac {n -1}{n}}}{a -b}-\left (\int f \left (x \right )d x \right )-c_{1} = 0 \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (y-a \right )^{-\frac {n +1}{n}} \left (y-b \right )^{-\frac {n -1}{n}}}{a -b}-\left (\int f \left (x \right )d x \right )-c_{1} &= 0 \\ \end{align*}

Verification of solutions

\[ -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (y-a \right )^{-\frac {n +1}{n}} \left (y-b \right )^{-\frac {n -1}{n}}}{a -b}-\left (\int f \left (x \right )d x \right )-c_{1} = 0 \] Verified OK.

1.548.2 Solving as first order nonlinear p but separable ode

The ode has the form \begin {align*} (y')^{\frac {n}{m}} &= f(x) g(y)\tag {1} \end {align*}

Where \(n=n, m=1, f=f \left (x \right )^{n} , g=\left (y -a \right )^{n +1} \left (y -b \right )^{n -1}\). Hence the ode is \begin {align*} (y')^{n} &= f \left (x \right )^{n} \left (y -a \right )^{n +1} \left (y -b \right )^{n -1} \end {align*}

Solving for \(y^{\prime }\) from (1) gives \begin {align*} y^{\prime } &={\mathrm e}^{\frac {\ln \left (f g \right )}{n}} \end {align*}

To be able to solve as separable ode, we have to now assume that \(f>0,g>0\). \begin {align*} f \left (x \right )^{n} &> 0\\ \left (y -a \right )^{n +1} \left (y -b \right )^{n -1} &> 0 \end {align*}

Under the above assumption the differential equations become separable and can be written as \begin {align*} y^{\prime } &=f^{\frac {1}{n}} g^{\frac {1}{n}} \end {align*}

Therefore \begin {align*} g^{-\frac {1}{n}} \, dy &= \left (f^{\frac {1}{n}}\right )\,dx \end {align*}

Replacing \(f(x),g(y)\) by their values gives \begin {align*} \left (\left (y -a \right )^{n +1} \left (y -b \right )^{n -1}\right )^{-\frac {1}{n}} \, dy &= \left (\left (f \left (x \right )^{n}\right )^{\frac {1}{n}}\right )\,dx \end {align*}

Integrating now gives the solutions. \begin {align*} \int \left (\left (y -a \right )^{n +1} \left (y -b \right )^{n -1}\right )^{-\frac {1}{n}}d y &= \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \end {align*}

Integrating gives \begin {align*} -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (\left (y-a \right )^{n +1} \left (y-b \right )^{n -1}\right )^{-\frac {1}{n}}}{a -b} &= \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \end {align*}

Therefore \begin{align*} -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (\left (y-a \right )^{n +1} \left (y-b \right )^{n -1}\right )^{-\frac {1}{n}}}{a -b} &= \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (\left (y-a \right )^{n +1} \left (y-b \right )^{n -1}\right )^{-\frac {1}{n}}}{a -b} &= \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \\ \end{align*}

Verification of solutions

\[ -\frac {n \left (-y+a \right ) \left (-y+b \right ) \left (\left (y-a \right )^{n +1} \left (y-b \right )^{n -1}\right )^{-\frac {1}{n}}}{a -b} = \int \left (f \left (x \right )^{n}\right )^{\frac {1}{n}}d x +c_{1} \] Verified OK. {0 < f(x)^n, 0 < (y-a)^(n+1)*(y-b)^(n-1)}

1.548.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{n}-f \left (x \right )^{n} \left (y-a \right )^{n +1} \left (y-b \right )^{n -1}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{\frac {n \ln \left (f \left (x \right )\right )+\ln \left (y-a \right ) n +\ln \left (y-b \right ) n +\ln \left (y-a \right )-\ln \left (y-b \right )}{n}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (y-b \right )^{\frac {1}{n}}}{\left (y-a \right )^{\frac {1}{n}} \left (-y+b \right ) \left (-y+a \right )}=f \left (x \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (y-b \right )^{\frac {1}{n}}}{\left (y-a \right )^{\frac {1}{n}} \left (-y+b \right ) \left (-y+a \right )}d x =\int f \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\left (y-b \right )^{\frac {1}{n}} n}{\left (y-a \right )^{\frac {1}{n}} \left (a -b \right )}=\int f \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{\ln \left (-\frac {a c_{1} -c_{1} b +a \left (\int f \left (x \right )d x \right )-b \left (\int f \left (x \right )d x \right )}{n}\right ) n} a -b}{-1+{\mathrm e}^{\ln \left (-\frac {a c_{1} -c_{1} b +a \left (\int f \left (x \right )d x \right )-b \left (\int f \left (x \right )d x \right )}{n}\right ) n}} \end {array} \]

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 
trying simple symmetries for implicit equations 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.734 (sec). Leaf size: 55

dsolve(diff(y(x),x)^n-f(x)^n*(y(x)-a)^(n+1)*(y(x)-b)^(n-1)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {b \left (-\frac {n}{\left (a -b \right ) \left (\int f \left (x \right )d x +c_{1} \right )}\right )^{n}-a}{-1+\left (-\frac {n}{\left (a -b \right ) \left (\int f \left (x \right )d x +c_{1} \right )}\right )^{n}} \]

✓ Solution by Mathematica

Time used: 6.298 (sec). Leaf size: 79

DSolve[-(f[x]^n*(-a + y[x])^(1 + n)*(-b + y[x])^(-1 + n)) + y'[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {b n^n+a (a-b)^n \left (\int _1^x-(-1)^{\frac {1}{n}} f(K[1])dK[1]+c_1\right ){}^n}{n^n+(a-b)^n \left (\int _1^x-(-1)^{\frac {1}{n}} f(K[1])dK[1]+c_1\right ){}^n} \]