In[]:=
ClearAll["Global'*"]​​Remove[x]​​​​df[x_[t_]]=4*D[x[t],{t,2}]+2*D[x[t],{t,1}]+4*D[x[t],{t,0}]​​​​ih[t_]=-9E^(-t/4)Sin[t(Sqrt[15]/4)]​​​​​​df1s=FullSimplify[DSolve[df[x[t]]==ih[t],x,t]];​​Evaluate[x[t]/.df1s]​​​​​​sln={{x->Function[{t},(C[2]*Cos[(Sqrt[15]*t)/4])/E^(t/4)+(C[1]*Sin[(Sqrt[15]*t)/4])/E^(t/4)+​​((3/10)*(Sqrt[15]*t*Cos[(Sqrt[15]*t)/4]))/E^(t/4)]}};​​Evaluate[x[t]/.sln]​​​​Simplify[df[x[t]]/.sln]​​Simplify[df[x[t]]/.df1s]​​​​WolframAlpha["DSolve[4* D[x[t],{t,2 }] + 2* D[x[t],{t,1 }] + 4* D[x[t],{t,0 }]== -9 E^(-t/4) Sin[t(Sqrt[15]/4)],x,t]"]​​​​​​
Out[]=
4x[t]+2
′
x
[t]+4
′′
x
[t]
Out[]=
-9
-t/4

Sin
15
t
4

Out[]=

-t/4


2
Cos
15
t
4
+
-t/4


1
Sin
15
t
4
+
3
10
-t/4

15
tCos
15
t
4
+2Cos
15
t
2
Sin
15
t
4
-2Cos
15
t
4
Sin
15
t
2

Out[]=

3
2
3
5
-t/4

tCos
15
t
4
+
-t/4


2
Cos
15
t
4
+
-t/4


1
Sin
15
t
4

Out[]=
-9
-t/4

Sin
15
t
4

Out[]=
-9
-t/4

Sin
15
t
4

Out[]=
Input
DSolve4×
2
∂
x(t)
∂
2
t
+2×
∂x(t)
∂
1
t
+4×
0
∂
x(t)
∂
0
t
-9
-t/4

sint×
15
4
,x,t
ODE classification
second-order linear ordinary differential equation
Differential equation solution
Approximate form
☑ Step-by-step solution
x(t)
c
1
-t/4

sin
15
t
4
+
c
2
-t/4

cos
15
t
4
+
3
2
3
5
-t/4

tcos
15
t
4
Plots of sample individual solutions
x(0)1
′
x
(0)0
x(0)0
′
x
(0)1
Sample solution family
(sampling x(0) and
′
x
(0))
Interactive differential equation solution plots
x(t)
Initial conditions:
x(0)
′
x
(0)
More controls
Possible Lagrangian
ℒ(
′
x
,x,t)
1
2
-
t/2

2
x
+
t/2

2
(
′
x
)
-
9
2
t/4

xsin
15
t
4
​
​