60.4.19 problem 1467

Internal problem ID [11470]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1467
Date solved : Monday, January 27, 2025 at 11:22:28 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+\operatorname {a2} y^{\prime \prime }+\operatorname {a1} y^{\prime }+\operatorname {a0} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 585

dsolve(diff(diff(diff(y(x),x),x),x)+a2*diff(diff(y(x),x),x)+a1*diff(y(x),x)+a0*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{-\frac {x \left (\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{2}/{3}}+\frac {\operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}}{3}+\left (i \sqrt {3}-1\right ) \left (\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}\right )\right )}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{2}/{3}} \sqrt {3}-4 i \sqrt {3}\, \operatorname {a2}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{2}/{3}}-4 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}-4 \operatorname {a2}^{2}+12 \operatorname {a1} \right ) x}{12 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}}}+c_3 \,{\mathrm e}^{\frac {\left (\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{2}/{3}}-2 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}+4 \operatorname {a2}^{2}-12 \operatorname {a1} \right ) x}{6 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{{1}/{3}}}} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 84

DSolve[a0*y[x] + a1*D[y[x],x] + a2*D[y[x],{x,2}] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]}+c_2 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}+c_3 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]} \]