Hello jerome,
1.) We are given to solve the inhomogeneous linear 3rd order ODE:
(1) $$y'''+y'=4\sin(x)+2x^2$$
The differential operator $$D^2+1$$ annihilates $$4\sin(x)$$ and $$D^3$$ annihilates $$2x^2$$ hence the operator:
$$A\equiv D^3(D^2+1)$$
annihilates $$4\sin(x)+2x^2$$.
Thus, applying $$A$$ to both sides of (1) gives us:
(2) $$D^4(D^2+1)^2[y]=0$$
The characteristic roots are then:
$$r=0,\,\pm i$$
where $$r=0$$ is of multiplicity 4 and $$r=\pm i$$ are of multiplicity 2, and so the general solution to (2) is given by:
(3) $$y(x)=c_1+c_2x+c_3x^2+c_4x^3+(c_5+c_6x)\cos(x)+(c_7+c_8x)\sin(x)$$
Now, recall that a general solution to (1) is of the form $$y(x)=y_h(x)+y_p(x)$$. Since every solution to (1) is also a solution to (2), then $$y(x)$$ must have the form displayed on the right-hand side of (3). But we recognize that:
$$y_h(x)=c_1+c_2\cos(x)+c_3\sin(x)$$
and so there must exist a particular solution of the form:
$$y_p(x)=c_2x+c_3x^2+c_4x^3+c_6x\cos(x)+c_8x\sin(x)$$
Now it is a matter of using the method of undetermined coefficients to determine the particular solution that satisfies (1).
So, what we need to do is compute $$y_p'(x)$$ and $$y_p'''(x)$$, substitute these into (1), and solve the resulting linear system that arises when we equate coefficients. So, computing the necessary derivatives, we find:
$$y_p'(x)=c_2+2c_3x+3c_4x^2+c_6(\cos(x)-x\sin(x))+c_8(\sin(x)+x\cos(x))$$
$$y_p''(x)=2c_3+6c_4x+c_6(-2\sin(x)-x\cos(x))+c_8(2\cos(x)-x\sin(x))$$
$$y_p'''(x)=6c_4+c_6(-3\cos(x)+x\sin(x))+c_8(-3\sin(x)-x\cos(x))$$
Substituting into (1), we find:
$$\left(6c_4+c_6(-3\cos(x)+x\sin(x))+c_8(-3\sin(x)-x\cos(x)) \right)+\left(c_2+2c_3x+3c_4x^2+c_6(\cos(x)-x\sin(x))+c_8(\sin(x)+x\cos(x)) \right)=4\sin(x)+2x^2$$
Rearranging, we may write:
$$-2c_6\cos(x)-2c_8\sin(x)+3c_4x^2+2c_3x+c_2+6c_4=0\cos(x)+4\sin(x)+2x^2+0x+0$$
Equating coefficients, we find:
$$c_6=0$$
$$c_8=-2$$
$$c_4=\frac{2}{3}$$
$$c_3=0$$
$$c_2=-4$$
Hence, we find:
$$y_p(x)=-4x+\frac{2}{3}x^3-2x\sin(x)$$
and by the principle of superposition, we find:
$$y(x)=c_1+c_2\cos(x)+c_3\sin(x)-4x+\frac{2}{3}x^3-2x\sin(x)$$
2.) We are given to solve the inhomogeneous linear 3rd order ODE:
(1) $$y'''+y''+3y'-5y=5\sin(2x)+10x^2-3x+7$$
The differential operator $$\frac{D^2}{2^2}+1$$ annihilates $$5\sin(2x)$$ and $$D^3$$ annihilates $$10x^2-3x+7$$ hence the operator:
$$A\equiv \frac{1}{4}D^3(D^2+4)$$
annihilates $$5\sin(2x)+10x^2-3x+7$$.
Thus, applying $$A$$ to both sides of (1) gives us:
(2) $$D^3(D^2+4)(D-1)(D^2+2D+5)[y]=0$$
The characteristic roots are then:
$$r=0,\pm2i,1,-1\pm2i$$
The root $$r=0$$ is of multiplicity 3, and so the general solution to (2) is given by:
$$y(x)=c_1+c_2x+c_3x^2+c_4\cos(2x)+c_5\sin(2x)+c_6e^x+c_7e^{-x}\cos(2x)+c_8e^{-x}\sin(2x)$$
Now, recall that a general solution to (1) is of the form $$y(x)=y_h(x)+y_p(x)$$. Since every solution to (1) is also a solution to (2), then $$y(x)$$ must have the form displayed on the right-hand side of (3). But we recognize that:
$$y_h(x)=c_6e^x+c_7e^{-x}\cos(2x)+c_8e^{-x}\sin(2x)$$
and so there must exist a particular solution of the form:
$$y_p(x)=c_1+c_2x+c_3x^2+c_4\cos(2x)+c_5\sin(2x)$$
Now it is a matter of using the method of undetermined coefficients to determine the particular solution that satisfies (1).
So, what we need to do is compute $$y_p'(x),\,y_p''(x),\,y_p..'(x)$$ and substitute these into (1), and solve the resulting linear system that arises when we equate coefficients. So, computing the necessary derivatives, we find:
$$y_p'(x)=c_2+2c_3x-2c_4\sin(2x)+2c_5\cos(2x)$$
$$y_p''(x)=2c_3-4c_4\cos(2x)-4c_5\sin(2x)$$
$$y_p'''(x)=8c_4\sin(2x)-8c_5\cos(2x)$$
Substituting into (1), we find:
$$\left(8c_4\sin(2x)-8c_5\cos(2x) \right)+\left(2c_3-4c_4\cos(2x)-4c_5\sin(2x) \right)+3\left(c_2+2c_3x-2c_4\sin(2x)+2c_5\cos(2x) \right)-$$
$$\,\,\,\,\,\,\,\,5\left(c_1+c_2x+c_3x^2+c_4\cos(2x)+c_5\sin(2x) \right)=5\sin(2x)+10x^2-3x+7$$
Rearranging, we may write:
$$(-9c_4-2c_5)\cos(2x)+(2c_4-9c_5)\sin(2x)+(-5c_3)x^2+(-5c_2+6c_3)x+(-5c_1+3c_2+2c_3)=0\cos(2x)+5\sin(2x)+10x^2-3x+7$$
Equating coefficients, we find:
$$-9c_4-2c_5=0$$
$$2c_4-9c_5=5$$
$$-5c_3=10$$
$$-5c_2+6c_3=-3$$
$$-5c_1+3c_2+2c_3=7$$
Solving this system, we find:
$$c_1=-\frac{82}{25},\,c_2=-\frac{9}{5},\,c_3=-2,\,c_4=\frac{2}{17},\,c_5=-\frac{9}{17}$$
Hence, we find:
$$y_p(x)=-\frac{82}{25}-\frac{9}{5}x-2x^2+\frac{2}{17}\cos(2x)-\frac{9}{17}\sin(2x)$$
and by the principle of superposition, we find:
$$y(x)=c_1e^x+c_2e^{-x}\cos(2x)+c_3e^{-x}\sin(2x)-\frac{82}{25}-\frac{9}{5}x-2x^2+\frac{2}{17}\cos(2x)-\frac{9}{17}\sin(2x)$$
To jerome and any other guests viewing this topic, I invite and encourage you to post other differential equations problems in our http://www.mathhelpboards.com/f17/ forum.
Best Regards,
Mark.