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 Wroian of y_1 and y_2. Find p(x), q(x) and the general form of y_1 and y_2.

Second order linear equations
asked 2020-11-09
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 \(y_2\).

Answers (1)

\(y''+p(x)y'+q(x)=0\) (*)
Since \(y_1\), and \(y_2\) are solution to (*)
So their linear combinations are also a solution to (*)
Hence \(y_1+y_2=e^{-x}\) is also a solution to (*)
So, from (*) we have
\(\Rightarrow e^{-x}-e^{-x}p(x)+e^{-x}q(x)=0\)
or \(e^{-x}[1-p(x)+q(x)]=0\)
As \(e^{-x}\neq0\ \forall\ x\)
Hence, \(1-p(x)+q(x)=0\) (I)
Now, \(W[y_1,y_2]=[(y_1,y_2),(y'_1,y''_2)]=y_1y'_2-y'_1y_2=e^x\) (II)
As \(y_1+y_2=e^{-x}\)
\(\Rightarrow y_2=e^{-x}-y_1\)
and \(y'_1+y'_2=-e^{-x}\)
so, \(y'_2=e^{-x}-y'_1\)
Putting the value of \(y_2\) and \(y'_2\) in (II) we got,
\(\Rightarrow y'_1=-e^{2x}-y_1\)
\(\Rightarrow y''_1=-2e^{2x}-y_1\)
\(\Rightarrow y''_1=-2e^{2x}-y'_1=-2e^{2x}-(-e^{2x}-y_1)\)
Putting the value of \(y'_1\) and \(y''_1\) in (*) we get,
From (I) we have \(1-p(x)+q(x)=0\)
\(\Rightarrow e^{2x}[1+p(x)]=0\)
From (I) we have
\(\Rightarrow q(x)=-2\)
So, \(p(x)=-1\) and \(q(x)=-2\)
Hence the differential equation become,
The characteristic equation is
or \(v^2-2v+v-2=0\)
so, \((v-2)(v+1)=0\)
\(\Rightarrow v=2\) or \(v=-1\)
Hence \(y_1(x)=e^{2x}\) and \(y_2=e^{-x}\)
Here \(y(x)=c_1e^{2x}+c_2e^{-x}\)
Answer: \(p(x)=-1,\ q(x)=-2,\ y(x)=c_1e^{2x}+c_2e^{-x}\)
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-06-10
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
asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
asked 2021-01-28

One solution of the differential equation \(y" – 4y = 0\) is \(y = e^{2x} \)Find a second linearly independent solution using reduction of order.

asked 2021-02-12
\(y=3e^{3x}\) 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.
asked 2021-02-09
Solve the following non-homogeneous second order linear differential equation.
if \(p=4,\ q=8,\ r=7\).
asked 2021-02-02
Find the general solution of the second order non-homogeneous linear equation:
\(y''-7y'+12y=10\sin t+12t+5\)
asked 2021-01-04
Verify that the given functions form a basis of solutions of the given equation and solve the given initial value problem.
\(4x^2-3y=0,\ y(1)=3,\ y'(1)=2.5,\) the basis of solution are \(y_1=x^{-\frac{1}{2}}\) and \(y_2=x(\frac{3}{2})\)
asked 2021-03-25
Find a general solution to \(\displaystyle{y}{''}+{4}{y}'+{3.75}{y}={109}{\cos{{5}}}{x}\)
To solve this, the first thing I did was find the general solutionto the homogeneous equivalent, and got
Then i used the form \(\displaystyle{K}{\cos{{\left({w}{x}\right)}}}+{M}{\sin{{\left({w}{x}\right)}}}\) and got \(\displaystyle-{2.72}{\cos{{\left({5}{x}\right)}}}+{2.56}{\sin{{\left({5}{x}\right)}}}\) as a solution of the nonhomogeneous ODE
asked 2021-06-09
Change from rectangular to cylindrical coordinates. (Let \(r\geq0\) and \(0\leq\theta\leq2\pi\).)
a) \((-2, 2, 2)\)
b) \((-9,9\sqrt{3,6})\)
c) Use cylindrical coordinates.
where E is enclosed by the planes \(z=0\) and
and by the cylinders
\(x^{2}+y^{2}=16\) and \(x^{2}+y^{2}=36\)
d) Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone
and the sphere
asked 2021-05-17
Find equations of both lines through the point (2, ?3) that are tangent to the parabola \(y = x^2 + x\).
\(y_1\)=(smaller slope quation)
\(y_2\)=(larger slope equation)