29.5.19 problem 135

Internal problem ID [4736]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 5
Problem number : 135
Date solved : Monday, January 27, 2025 at 09:34:53 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=f \left (a +b x +c y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 39

dsolve(diff(y(x),x) = f(a+b*x+c*y(x)),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_a} c +a \right ) c +b}d \textit {\_a} \right ) c -x +c_{1} \right ) c -b x}{c} \]

✓ Solution by Mathematica

Time used: 0.223 (sec). Leaf size: 262

DSolve[D[y[x],x]==f[a+b x +c y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {f(a+b x+c K[2]) \int _1^x\left (\frac {c^2 f''(a+b K[1]+c K[2])}{b+c f(a+b K[1]+c K[2])}-\frac {c^3 f(a+b K[1]+c K[2]) f''(a+b K[1]+c K[2])}{(b+c f(a+b K[1]+c K[2]))^2}\right )dK[1] c+c+b \int _1^x\left (\frac {c^2 f''(a+b K[1]+c K[2])}{b+c f(a+b K[1]+c K[2])}-\frac {c^3 f(a+b K[1]+c K[2]) f''(a+b K[1]+c K[2])}{(b+c f(a+b K[1]+c K[2]))^2}\right )dK[1]}{b+c f(a+b x+c K[2])}dK[2]+\int _1^x\frac {c f(a+b K[1]+c y(x))}{b+c f(a+b K[1]+c y(x))}dK[1]=c_1,y(x)\right ] \]