# Solve the following non-homogeneous second order linear differential equation. y''+py'-qy=-rt if p=4, q=8, r=7.

Question
Solve the following non-homogeneous second order linear differential equation.
$$y''+py'-qy=-rt$$
if $$p=4,\ q=8,\ r=7$$.

2021-02-10
Substituting $$p=4,\ q=8,\ r=7$$ in the given equation: $$y''+4y'-8e=-7t$$. Thus, auxiliary equation of $$y''+4y'-8y=-7t$$ is $$m^2+4m-8=0$$. Solving for m:
$$m=\frac{4\pm\sqrt{4^2-4(-8)}}{2}$$
$$=-2\pm2\sqrt3$$
Thus the complementary function is $$y_CF=e^{-2t}(c_1e^{2\sqrt3t}+c_2e^{-2\sqrt3t})$$ where $$c_1$$ and $$c_2$$ are arbitrary constants.
Now to find Particular integral:
Using the operator $$D=\frac{d}{dt}$$ in $$y''+4y'-8y=-7t$$
$$(D^2+4D-8)y=-7t$$. Thus
$$y_{PI}=\frac{1}{D^2+4D-8}7t$$
$$=\frac{7}{8}\cdot\frac{1}{1-(\frac{D^2+4D}{8})}(t)$$
Now using binomial expansion: $$y_{PI}=\frac{7}{8}\cdot\frac{1}{1-(\frac{D^2+4D}{8})}(t)$$
$$=\frac{7}{8}\cdot(1+\frac{D^2+4D}{8}+(\frac{D^2+4D}{8})^2+...)(t)$$
$$=\frac{7}{8}(t+(\frac{D^2+4D}{8})t)$$
$$=\frac{7}{8}(t+\frac{1}{2})$$
$$=\frac{7t}{8}+\frac{7}{16}$$
Thus complete solution is $$y_{CF}+y_{PI}$$ or the complete solution is
$$e^{-2t}c_1e^{2\sqrt3t}+c_2e^{-2\sqrt3t}+\frac{7t}{8}+\frac{7}{16}$$ where $$c_1$$ and $$c_2$$ are arbitrary constant

### Relevant Questions

Let $$y_1$$ and $$y_2$$ be solution of a second order homogeneous linear differential equation $$y''+p(x)y'+q(x)=0$$, in R. Suppose that $$y_1(x)+y_2(x)=e^{-x}$$,
$$W[y_1(x),y_2(x)]=e^x$$, where $$W[y_1,y_2]$$ is the Wro
ian of $$y_1$$ and $$y_2$$.
Find p(x), q(x) and the general form of $$y_1$$ and $$y_2$$.
Find the general solution of the second order non-homogeneous linear equation:
$$y''-7y'+12y=10\sin t+12t+5$$
Determine whether the given set S is a subspace of the vector space V.
A. V=$$P_5$$, and S is the subset of $$P_5$$ consisting of those polynomials satisfying p(1)>p(0).
B. $$V=R_3$$, and S is the set of vectors $$(x_1,x_2,x_3)$$ in V satisfying $$x_1-6x_2+x_3=5$$.
C. $$V=R^n$$, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=$$C^2(I)$$, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=$$P_n$$, and S is the subset of $$P_n$$ consisting of those polynomials satisfying p(0)=0.
G. $$V=M_n(R)$$, and S is the subset of all symmetric matrices
$$y=3e^{3x}$$ is a solution of a second order linear homogeneous differential equation with constant coefficients. The equation is:
(a) $$y''-(3+a)y'+3ay=0$$, a any real number.
(b) $$y''+y'-6y=0$$
(c) $$y''+3y'=0$$
(d) $$y''+(3-a)y'+3ay=0$$, a any real number.
(e) Cannot be determined.
A dynamic system is represented by a second order linear differential equation.
$$2\frac{d^2x}{dt^2}+5\frac{dx}{dt}-3x=0$$
The initial conditions are given as:
when $$t=0,\ x=4$$ and $$\frac{dx}{dt}=9$$
Solve the differential equation and obtain the output of the system x(t) as afunction of t.
Solve the second order linear differential equation using method of undetermined coefficients
$$3y''+2y'-y=x^2+1$$
$$y''-4y'+9y=0,\ \ y(0)=0,\ \ y'(0)=-8$$
$$t(t^2-4)y''-ty'+3t^2y=0, y(1)=1 y'(1)=3$$
$$(\sin 0)y'''-(\cos 0)y'=2$$
One solution of the differential equation $$y" – 4y = 0$$ is \(y = e^{2x} Find a second linearly independent solution using reduction of order.