Internal problem ID [9086]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1507.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _exact, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (x a +b \right ) x y^{\prime \prime \prime }+\left (\alpha x +\beta \right ) y^{\prime \prime }+y^{\prime } x +y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1422
dsolve((a*x+b)*x*diff(diff(diff(y(x),x),x),x)+(alpha*x+beta)*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x)-f(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.618 (sec). Leaf size: 1650
DSolve[-f[x] + y[x] + x*y'[x] + (\[Beta] + \[Alpha]*x)*y''[x] + x*(b + a*x)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{-\frac {\beta }{b}} (b+a x)^{-\frac {\alpha }{a}+\frac {\beta }{b}+2} \left (\text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a x}{b}\right ] \left (c_1+\int _1^x\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[2]}{b}\right ]dK[2] \left (c_3+\int _1^x-\frac {b f(K[5]) (b+a K[5])^{\frac {\alpha }{a}-\frac {\beta }{b}-3}}{\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] \left ((2 b-\beta ) \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ]+a \text {HeunCPrime}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] K[5]\right )-a \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] \text {HeunCPrime}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] K[5]}dK[5]\right )+\int _1^x\frac {b f(K[3]) (b+a K[3])^{\frac {\alpha }{a}-\frac {\beta }{b}-3} \int _1^{K[3]}\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[2]}{b}\right ]dK[2]}{\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[3]}{b}\right ] \left ((2 b-\beta ) \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[3]}{b}\right ]+a \text {HeunCPrime}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[3]}{b}\right ] K[3]\right )-a \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[3]}{b}\right ] \text {HeunCPrime}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[3]}{b}\right ] K[3]}dK[3]\right ) x^{\frac {\beta }{b}}+\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a x}{b}\right ] \left (c_2-\int _1^x\text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[1]}{b}\right ] K[1]^{\frac {\beta }{b}-2}dK[1] \left (c_3+\int _1^x-\frac {b f(K[5]) (b+a K[5])^{\frac {\alpha }{a}-\frac {\beta }{b}-3}}{\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] \left ((2 b-\beta ) \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ]+a \text {HeunCPrime}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] K[5]\right )-a \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] \text {HeunCPrime}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[5]}{b}\right ] K[5]}dK[5]\right )+\int _1^x-\frac {b f(K[4]) (b+a K[4])^{\frac {\alpha }{a}-\frac {\beta }{b}-3} \int _1^{K[4]}\text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[1]}{b}\right ] K[1]^{\frac {\beta }{b}-2}dK[1]}{\text {HeunC}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[4]}{b}\right ] \left ((2 b-\beta ) \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[4]}{b}\right ]+a \text {HeunCPrime}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[4]}{b}\right ] K[4]\right )-a \text {HeunC}\left [\frac {b \alpha \beta -a \beta (b+\beta )}{a b^2},-\frac {b}{a^2},\frac {\beta }{b}-1,-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[4]}{b}\right ] \text {HeunCPrime}\left [\frac {2 \alpha }{a}-6,-\frac {b}{a^2},3-\frac {\beta }{b},-\frac {\alpha }{a}+\frac {\beta }{b}+3,0,-\frac {a K[4]}{b}\right ] K[4]}dK[4]\right ) x^2\right ) \\ \end{align*}