# On solution of the differetial equation y''+y'=0 is y=e^{-x}. Use Reduction of Order to find a second linearly independent solution. Question
Second order linear equations On solution of the differetial equation $$y''+y'=0$$ is $$y=e^{-x}$$. Use Reduction of Order to find a second linearly independent solution. 2021-02-13
In such problems ie second order linear differential equations when we are given one solution, $$y_1$$ we assume the second solution to be of the form, $$y_2=vy_1$$ and substitute $$y_2$$ in the given ode and reduce order of the differential equation by using the fact that $$y_1$$ is a solution.
Compute:$$y_2',y_2''$$
$$y_2'=vy"_1+v'y_1$$
$$=-ve^{-x}+v'e^{-x}$$
$$=e^{-x}(-v+v')$$
$$y_2''=-e^{-x}(-v+v')+e^{-x}(-v'+v'')$$
$$=e^{-x}(v-v'-v'+v'')$$
Substitute $$y_2', y_2''$$ in given differential equation
$$e^{-x}(v-2v'+v'')+e^{-x}(-v+v')=0$$
$$e^{-x}(-v'+v'')=0$$
$$-v'+v''=0$$
Let $$u=v'$$ we get a first order differential equation
\u+u'=0\)
$$u'=u'$$
$$u=e^x$$
Substitute $$u=v'$$ in above equation and solve for v
$$v'=e^x$$
Integrating we get
$$v=e^x$$
Get second linearly independent solution by substituting v in expression for $$y_2$$
$$y_2=ve^{-x}$$
$$=ce^xe^{-x}$$
\=c\)
Hence second linearly independent solution is c

### Relevant Questions One solution of the differential equation $$y" – 4y = 0$$ is $$y = e^{2x} Find a second linearly independent solution using reduction of order. asked 2021-02-05 A differential equation and a nontrivial solution f are given below. Find a second linearly independent solution using reduction of order. Assume that all constants of integration are zero. \(tx''-(2t+1)x+2x=0,\ t>0,\ f(t)=3*e^{2t}$$ Let $$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$$. $$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$$, a any real number.
(e) Cannot be determined. The function y=x is one the solution of
$$(x-1)y''-xy'+y=0$$
Solve using the reduction method to order Using the existence and uniqueness theorem for second order linear ordinary differential equations, find the largest interval in which the solution to the initial value is certain to exist.
$$t(t^2-4)y''-ty'+3t^2y=0, y(1)=1 y'(1)=3$$ Find the general solution of the second order non-homogeneous linear equation:
$$y''-7y'+12y=10\sin t+12t+5$$ given $$\displaystyle{y}=\frac{{1}}{{x}}$$ is a solution $$\displaystyle{2}{x}^{{2}}{d}{2}\frac{{y}}{{\left.{d}{x}\right.}}+{x}\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}-{3}{y}={0},{x}{>}{0}$$
a) Find a linearly independent solution by reduction the order approach
b) Show that 2 solutions are linearly independent
c) Write a general solution $$2\frac{d^2x}{dt^2}+5\frac{dx}{dt}-3x=0$$
when $$t=0,\ x=4$$ and $$\frac{dx}{dt}=9$$ Linear equations of second order with constant coefficients. Find all solutions on $$\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}+{4}{y}={0}$$