ODE
\[ y'''(x)+y''(x)+2 y'(x)+4 y(x)=\sin (2 x) \] ODE Classification
[[_3rd_order, _linear, _nonhomogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.396818 (sec), leaf count = 1183
\[\left \{\left \{y(x)\to e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]} c_1+e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]} c_2+e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]} c_3+\frac {8 \left (332-2 \left (64+118 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+3 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+2 \left (-84-15 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+5 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ] \left (-252+\left (-70+26 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right )+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]^2 \left (-26-3 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+18 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ] \left (10+25 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+8 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )\right )\right ) \cos (2 x)+\left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ] \left (128+24 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]-5 \left (-24-50 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+3 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )+4 \left (24+\left (116-30 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+23 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+\left (36+6 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]+11 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]^2 \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ] \left (-88+96 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+25 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )+4 \left (52+25 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]+51 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )\right )\right ) \sin (2 x)}{\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]^2 \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ] \left (4+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]^2\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right ) \left ((1-2 i)+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right ) \left ((1+2 i)+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right ) \left (-2+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,2\right ]^2+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]\right ) \left (4+\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2+2 \text {$\#$1}+4\& ,3\right ]^2\right )}\right \}\right \}\]
Maple ✓
cpu = 2.174 (sec), leaf count = 8292
\[\text {Expression too large to display}\] Mathematica raw input
DSolve[4*y[x] + 2*y'[x] + y''[x] + y'''[x] == Sin[2*x],y[x],x]
Mathematica raw output
{{y[x] -> E^(x*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0])*C[1] + E^(x*Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0])*C[2] + E^(x*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*C[3] + (8*Cos[2*x]*(332 - 2*(64 + 118*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + 3*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + 2*(-84 - 15*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + 5*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2 + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]*(-252 + (-70 + 26*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]) + 2*Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]^2*(-26 - 3*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + 18*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2 + Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]*(10 + 25*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + 8*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2))) + (Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]*(128 + 24*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] - 5*(-24 - 50*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + 3*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2) + 4*(24 + (116 - 30*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + 23*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + (36 + 6*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] + 11*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2)*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2) + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]^2*(Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]*(-88 + 96*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + 25*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2) + 4*(52 + 25*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0] + 51*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2)))*Sin[2*x])/(Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]^2*(Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] - Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0])*Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]*(4 + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]^2)*(Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0] - Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*(Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0] - Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*((1 - 2*I) + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0] + Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*((1 + 2*I) + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0] + Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*(-2 + Root[4 + 2*#1 + #1^2 + #1^3 & , 2, 0]^2 + 2*Root[4 + 2*#1 + #1^2 + #1^3 & , 1, 0]*Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0])*(4 + Root[4 + 2*#1 + #1^2 + #1^3 & , 3, 0]^2))}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)+2*diff(y(x),x)+4*y(x) = sin(2*x), y(x))
Maple raw output
[y(x) = 1/166*(4357*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(4/3)*83^(1/2)*sin(2*x)*exp(-1/75*x*(3*3^(1/2)*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(2/3)-3*249^(1/2)*(46+3*249^(1/2))^(2/3)+46*(46+3*249^(1/2))^(2/3)-46*(46+3*3^(1/2)*83^(1/2))^(2/3)+25*(46+3*249^(1/2))^(1/3)-25*(46+3*3^(1/2)*83^(1/2))^(1/3)))-45816*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(4/3)*3^(1/2)*exp(-1/150*(3*249^(1/2)*(46+3*249^(1/2))^(2/3)-46*(46+3*249^(1/2))^(2/3)-25*(46+3*249^(1/2))^(1/3)+50)*x)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*Int(-1/12450*83^(1/2)*sin(2*x)*(9*(46+3*249^(1/2))^(2/3)*249^(1/2)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)-9*83^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(2/3)+46*3^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(2/3)-138*(46+3*249^(1/2))^(2/3)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)-25*3^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(1/3)-75*(46+3*249^(1/2))^(1/3)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x))*exp(1/150*(3*249^(1/2)*(46+3*249^(1/2))^(2/3)-46*(46+3*249^(1/2))^(2/3)-25*(46+3*249^(1/2))^(1/3)+50)*x),x)+45816*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(4/3)*3^(1/2)*exp(-1/150*(3*249^(1/2)*(46+3*249^(1/2))^(2/3)-46*(46+3*249^(1/2))^(2/3)-25*(46+3*249^(1/2))^(1/3)+50)*x)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*Int(-1/12450*83^(1/2)*sin(2*x)*(9*(46+3*249^(1/2))^(2/3)*249^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)+9*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(2/3)-46*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(2/3)*3^(1/2)-138*(46+3*249^(1/2))^(2/3)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)+25*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)-75*(46+3*249^(1/2))^(1/3)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x))*exp(1/150*(3*249^(1/2)*(46+3*249^(1/2))^(2/3)-46*(46+3*249^(1/2))^(2/3)-25*(46+3*249^(1/2))^(1/3)+50)*x),x)+525*sin(2*x)*3^(1/2)*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(2/3)*exp(-1/75*x*(3*3^(1/2)*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(2/3)-3*249^(1/2)*(46+3*249^(1/2))^(2/3)+46*(46+3*249^(1/2))^(2/3)-46*(46+3*3^(1/2)*83^(1/2))^(2/3)+25*(46+3*249^(1/2))^(1/3)-25*(46+3*3^(1/2)*83^(1/2))^(1/3)))-68724*(46+3*3^(1/2)*83^(1/2))^(4/3)*sin(2*x)*exp(-1/75*x*(3*3^(1/2)*83^(1/2)*(46+3*3^(1/2)*83^(1/2))^(2/3)-3*249^(1/2)*(46+3*249^(1/2))^(2/3)+46*(46+3*249^(1/2))^(2/3)-46*(46+3*3^(1/2)*83^(1/2))^(2/3)+25*(46+3*249^(1/2))^(1/3)-25*(46+3*3^(1/2)*83^(1/2))^(1/3)))+2801250*Int(-1/12450*83^(1/2)*sin(2*x)*(9*(46+3*249^(1/2))^(2/3)*249^(1/2)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)-9*83^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(2/3)+46*3^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(2/3)-138*(46+3*249^(1/2))^(2/3)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)-25*3^(1/2)*cos(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x)*(46+3*3^(1/2)*83^(1/2))^(1/3)-75*(46+3*249^(1/2))^(1/3)*sin(1/150*(46+3*3^(1/2)*83^(1/2))^(1/3)*(9*(46+3*3^(1/2)*83^(1/2))^(1/3)*83^(1/2)-46*3^(1/2)*(46+3*3^(1/2)*83^(1/2))^(1/3)+25*3^(1/2))*x))*exp(1/150*(3*249^(1/2)*(46+3*249^(1/2))^(2/3)-46*(46+3*249^(1/2))^(2/3)-25*(46+3*249^(1/2))^(1/3)+50)*x),x)*exp(-1/150*(3*249^(1/2)*(46+3*249^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