Variable Separable

  • Some differential equations can be solved by the method of separation of variables.
  • (dy/dx) = f(x, y) is a first order and first degree differential equation. If f(x, y) can be written as h(x).g(y), where h(x) is a function of x and g(y) is a function of y, then we can separate h(x) and g(y).
  • Solutions of differential equations of the type (dy/dx) = f(ax + by + c):

Step1: Substitute ax + by + c = v in the given differential equation and change the differential coefficients accordingly.

Step2: Obtain the relation f(v)dv = g(x)dx

Step3: Integrate the obtained equation to get a relation between x, v and arbitrary constants.

Step4: Substitute the value of v in the relation obtained in step-3 to get a relation between x, y and arbitrary constants.

Keywords: Variable Separable method, Solutions of differential equations

 

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  • Q1

    Marks:1
    Answer:

    Explanation:

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  • Q2

    Marks:1
    Answer:

    Explanation:
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  • Q3

    The general solution of the differential equation

    (1 + x2) dy = (1 + y2) dx is

    Marks:1
    Answer:

    tan-1y = tan-1x + C.

    Explanation:
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  • Q4

    The general solution of the differential equation y(x2y + ex)dx – exdy = 0 is

    Marks:1
    Answer:

    x3y + 3ex = Cy

    Explanation:

    We have, yx2y+exdx-exdy=0

    ex dydx=x2y2+yex

    Dividing  by y2ex, we get

    1y2 dydx-1y=x2e-x

    Putting 1y=v, we get

    -1y2 dydx=dvdx

    Thus, we have  dydx+v=-x2e-x which is a linear differential equation.

    I.F.= e1dx=ex

    Hence, the solution is

    v·ex=-x2e-x·exdx+C3

    or  1yex=-x33+C3x3y+3ex=Cy

     

     

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  • Q5

    The differential equation, (ex + 1)ydy = (y + 1)exdx has the solution

    Marks:1
    Answer:

    (1 + y)(ex + 1) = Cey.

    Explanation:

    View Answer