CLASS XII DIFFERENTIAL EQUATION
DIFFERENTIAL EQUATION
1. ORDER of differential equation: It is the order of the highest order
derivative appearing in given differential equation.
Ex. d²y / dx²+ 2y dy/ dx + 5 = 0 has order = 2
2. Degree : It is the degree of highest order derivative but differential
coefficients should be free from p/q (Radical)
Ex. (dy/dx+y) 3/2 = d2y/dx2 Remove 3/2 by squaring both sides
So, (dy/dx + y)3 = (d2y/dx2)² so order = 2, degree = 2
Note: we can't find degree, If dy/dx act as Angle (Sin(dy/dx) etc) or
in exponent of edy/dx etc.
✤ TYPES OF DIFFERENTIAL EQUATIONS
① Variable separable solution of vs.:
1. Separate the function of with du and function of y
with dy.
2. Integrate both side
3. add 'c' on x side.
② Homogenius
Standard form
dy/dx = f(x,y) / g(x,y)
& degree of f(x,y) = degree g f(x,y)
Steps to check: Put x=kx and y = ky, then k should be cancel
out.
Steps to solve: 1. Put y=vx and dy/dx = v + x dv/dx
2. Apply the concept of variable seperable.
3. at last Put v = y/x.
③ Linear diff. Eq.
Standard form.
dy/dx + Py = Q.
Here P, Q are the function of x.
Ex. dy/dx + y/x = cosx
Steps to solve:
1. find Integrating factor I.f. = e∫Pdx
2. Sol. Is y e∫Pdx = ∫e∫Pdx . Q dx.
Note: emlogk = xm
2nd type homogeneous
dx/dy = f(x,y)/g(x,y)
steps to solve:
1. Put x=vy and dx/dy = V+y dv/dy
2. Apply the concept of variable seprable.
3. at last put v = x/y.
2nd type linear differential equation.
Dx/dy + Px = Q.
Here P, Q are the function of y
Ex: dx/dy + tany.x = y2+1/y
steps to solve:
1. find I.F. = e∫Pdy
2. Solution is
x.e∫Pdy = ∫e∫Pdy.Q dy.
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