Find maximal k such that $y''+ky=sin(x)cdotsin(6x)$ has no periodic solution.
I have the equation: $$y''+ky=sin(x)cdotsin(6x).tag{1}$$
I know that $$sin(x)cdotsin(6x)=frac{1}{2}(cos(5x)-cos(7x)).tag{2}$$
The homogeneous solution is $$y_h:=Asin(sqrt{k}x)+Bcos(sqrt{k}x).tag{3}$$
I tried to define $A:=A(x)$ and $B:=B(x)$ and differentiate twice. Then I substitute the $y''$ and $y$ in the equation. I obtained the next equation:
$$(A''(x)-2sqrt{x}B'(x))sin(sqrt{k}x)+(B''(x)+2sqrt{x}A'(x))cos(sqrt{k}x)=sin(x)cdotcos(x).tag{4}$$
How to find the maximal $k$ such that equation $(1)$ has no periodic solution?
Edit #1:
I've found the particular solution if $kneq 25$ and $kneq 49$.
$$Y_P(x)=Asin(5x)+Bcos(5x)+Csin(7x)+Dcos(7x)$$
where $A=0$, $B=frac{1}{2(k-25)}$, $C=0$, $D=frac{1}{2(k-49)}$.
There are solutions when $k=25$ and $k=49$, when we use particular solution of type $(Ax+B)sin(sqrt{k}x)+(Cx+D)cos(sqrt{k}x)+...$
I didn't understand the task "to find the greatest $k$ such that equation has no periodic solution". So now I understand they just meant the number $49$.
differential-equations
edited Dec 2 at 7:48
Tianlalu
3,06021038
asked Dec 1 at 10:44
ged
338113
add a comment |
I have the equation: $$y''+ky=sin(x)cdotsin(6x).tag{1}$$
I know that $$sin(x)cdotsin(6x)=frac{1}{2}(cos(5x)-cos(7x)).tag{2}$$
The homogeneous solution is $$y_h:=Asin(sqrt{k}x)+Bcos(sqrt{k}x).tag{3}$$
I tried to define $A:=A(x)$ and $B:=B(x)$ and differentiate twice. Then I substitute the $y''$ and $y$ in the equation. I obtained the next equation:
$$(A''(x)-2sqrt{x}B'(x))sin(sqrt{k}x)+(B''(x)+2sqrt{x}A'(x))cos(sqrt{k}x)=sin(x)cdotcos(x).tag{4}$$
How to find the maximal $k$ such that equation $(1)$ has no periodic solution?
Edit #1:
I've found the particular solution if $kneq 25$ and $kneq 49$.
$$Y_P(x)=Asin(5x)+Bcos(5x)+Csin(7x)+Dcos(7x)$$
where $A=0$, $B=frac{1}{2(k-25)}$, $C=0$, $D=frac{1}{2(k-49)}$.
There are solutions when $k=25$ and $k=49$, when we use particular solution of type $(Ax+B)sin(sqrt{k}x)+(Cx+D)cos(sqrt{k}x)+...$
I didn't understand the task "to find the greatest $k$ such that equation has no periodic solution". So now I understand they just meant the number $49$.
differential-equations
edited Dec 2 at 7:48
Tianlalu
3,06021038
asked Dec 1 at 10:44
ged
338113
4
You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38
add a comment |
I have the equation: $$y''+ky=sin(x)cdotsin(6x).tag{1}$$
I know that $$sin(x)cdotsin(6x)=frac{1}{2}(cos(5x)-cos(7x)).tag{2}$$
The homogeneous solution is $$y_h:=Asin(sqrt{k}x)+Bcos(sqrt{k}x).tag{3}$$
I tried to define $A:=A(x)$ and $B:=B(x)$ and differentiate twice. Then I substitute the $y''$ and $y$ in the equation. I obtained the next equation:
$$(A''(x)-2sqrt{x}B'(x))sin(sqrt{k}x)+(B''(x)+2sqrt{x}A'(x))cos(sqrt{k}x)=sin(x)cdotcos(x).tag{4}$$
How to find the maximal $k$ such that equation $(1)$ has no periodic solution?
Edit #1:
I've found the particular solution if $kneq 25$ and $kneq 49$.
$$Y_P(x)=Asin(5x)+Bcos(5x)+Csin(7x)+Dcos(7x)$$
where $A=0$, $B=frac{1}{2(k-25)}$, $C=0$, $D=frac{1}{2(k-49)}$.
There are solutions when $k=25$ and $k=49$, when we use particular solution of type $(Ax+B)sin(sqrt{k}x)+(Cx+D)cos(sqrt{k}x)+...$
I didn't understand the task "to find the greatest $k$ such that equation has no periodic solution". So now I understand they just meant the number $49$.
differential-equations
edited Dec 2 at 7:48
Tianlalu
3,06021038
asked Dec 1 at 10:44
ged
338113
I have the equation: $$y''+ky=sin(x)cdotsin(6x).tag{1}$$
I know that $$sin(x)cdotsin(6x)=frac{1}{2}(cos(5x)-cos(7x)).tag{2}$$
The homogeneous solution is $$y_h:=Asin(sqrt{k}x)+Bcos(sqrt{k}x).tag{3}$$
I tried to define $A:=A(x)$ and $B:=B(x)$ and differentiate twice. Then I substitute the $y''$ and $y$ in the equation. I obtained the next equation:
$$(A''(x)-2sqrt{x}B'(x))sin(sqrt{k}x)+(B''(x)+2sqrt{x}A'(x))cos(sqrt{k}x)=sin(x)cdotcos(x).tag{4}$$
How to find the maximal $k$ such that equation $(1)$ has no periodic solution?
Edit #1:
I've found the particular solution if $kneq 25$ and $kneq 49$.
$$Y_P(x)=Asin(5x)+Bcos(5x)+Csin(7x)+Dcos(7x)$$
where $A=0$, $B=frac{1}{2(k-25)}$, $C=0$, $D=frac{1}{2(k-49)}$.
There are solutions when $k=25$ and $k=49$, when we use particular solution of type $(Ax+B)sin(sqrt{k}x)+(Cx+D)cos(sqrt{k}x)+...$
I didn't understand the task "to find the greatest $k$ such that equation has no periodic solution". So now I understand they just meant the number $49$.
differential-equations
differential-equations
edited Dec 2 at 7:48
Tianlalu
3,06021038
asked Dec 1 at 10:44
ged
338113
edited Dec 2 at 7:48
Tianlalu
3,06021038
asked Dec 1 at 10:44
ged
338113
edited Dec 2 at 7:48
Tianlalu
3,06021038
edited Dec 2 at 7:48
Tianlalu
3,06021038
edited Dec 2 at 7:48
Tianlalu
3,06021038
3,06021038
asked Dec 1 at 10:44
ged
338113
asked Dec 1 at 10:44
ged
338113
asked Dec 1 at 10:44
ged
338113
338113
4
You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38
add a comment |
4
You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38
4
4
You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38
You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38
add a comment |
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You need to get a non-periodic particular solution, which only happens in the resonance cases $k=5^2$ and $k=7^2$.
– LutzL
Dec 1 at 11:38