Menu

Linear Equations With Constant Coefficients

अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients):

  • इस आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.Definition:A linear differential equation is an equation in which the dependent variable y
    and its differential coefficients occur only in the first degree. The general
    form of such an equation is
  • आपको यह जानकारी रोचक व ज्ञानवर्धक लगे तो अपने मित्रों के साथ इस गणित के आर्टिकल को शेयर करें ।यदि आप इस वेबसाइट पर पहली बार आए हैं तो वेबसाइट को फॉलो करें और ईमेल सब्सक्रिप्शन को भी फॉलो करें जिससे नए आर्टिकल का नोटिफिकेशन आपको मिल सके।यदि आर्टिकल पसन्द आए तो अपने मित्रों के साथ शेयर और लाईक करें जिससे वे भी लाभ उठाए।आपकी कोई समस्या हो या कोई सुझाव देना चाहते हैं तो कमेंट करके बताएं।इस आर्टिकल को पूरा पढ़ें।

Also Read This Article:mathematics and moving sofa

  • \left(\frac{d^{n}y}{dx^{n}}\right)+P_{1}\left(\frac{d^{n-1}y}{dx^{n-1}}\right)+P_{2}\left(\frac{d^{n-2}y}{dx^{n-2}}\right)+……..+P_{n}y=0         ……(i)
    Where Q
    and P_{1},P_{2},P_{3},…..,P_{n}are all
    constants or functions of x.

    If P_{1},P_{2},P_{3},…..,P_{n}
    are all constants (Q may not be constant), then the equation is said to be a linear differential equation
    constant coefficients.

 

  • We shall
    first of all consider the differential equation in which the second member viz
    Q is zero
  • i.e. \left(\frac{d^{n}y}{dx^{n}}\right)+P_{1}\left(\frac{d^{n-1}y}{dx^{n-1}}\right)+P_{2}\left(\frac{d^{n-2}y}{dx^{n-2}}\right)+……..+P_{n}y=0          ……(ii)
    \text{ If } y=f_1\left(x\right)
    be a solution of (ii),then by substitution in (ii) it can be seen that y=cf_{1}\left(x\right),where
    C is an arbitrary constant,is also a solution of (ii).

    Similarly
    if y=f_{2}\left(x\right),y=f_{3}\left(x\right)……..,y=f_{n}\left(x\right) are the
    solutions of (ii),then y=C_{2}f_{2}\left(x\right),y=C_{3}f_{3}\left(x\right),….,y=C_{n}f_{n}\left(x\right),
    where C_{2},C_{3},…..,C_{n} are arbitrary constants,are
    also the solutions of (ii) Also substitution will show that

    y=C_{1}f_{1}\left(x\right)+C_{2}f_{2}\left(x\right)+…+C_{n}f_{n}\left(x\right)                                      ………(iii)
    Is also a
    solution of (ii) \text{ If } f_{1}\left(x\right),f_{2}\left(x\right),f_{3}\left(x\right),….are
    linearly independent,then (iii) is the complete integral of (ii),since it
    contains n arbitrary constants and (ii) is order n.

    Now let
    us consider the equation (i),in which the second member viz. Q is also zero.If
    y=
    f(x)
    be solution of (i),then
    y=F(x)+f(x) be a solution of (i),then
    where F(x)= C_{1}f_{1}\left(x\right)+C_{2}f_{2}\left(x\right)+…+C_{n}f_{n}\left(x\right)is also a solution of (i) since the
    substitution of F(x) for y in the left hand member of (i) gives:zero and that of:f(x)
    for y gives Q,as y=f(x)
    is solution of (i).

Also Read This Article:student misconceptions about probability in coin flipping

  • solution (iv) contains n arbitrary constants and (i) is differential equation
    of nth order,therefore it is the complete solution of (i). The part F(x) is called the complementary function (C.F.) and the part f(x) is called the (P.I.).

Also Read This Article:Linear differential equation

  • उपर्युक्त आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.

Linear Equations With Constant Coefficients

अचर गुणांकों वाले रैखिक समीकरण
(Linear Equations With Constant Coefficients)

Linear Equations With Constant Coefficients

इस आर्टिकल में अचर गुणांकों वाले रैखिक समीकरण (Linear Equations With Constant Coefficients) के बारे में बताया गया है.
Definition:A linear differential equation is an equation in which the dependent variable y

No.Social MediaUrl
1.Facebookclick here
2.you tubeclick here
3.Instagramclick here
4.Linkedinclick here
5.Facebook Pageclick here

Leave a Reply

Your email address will not be published. Required fields are marked *