Differential Equations - 20 - Mathematics MCQ

The differential equation (i) is said to be homogeneous if the function f(x) is identically zero. The function f(x) is called the non-homogeneous term.

Discussion : The given differential equation is Dⁿy + a₁D^n-1y + a₂D^n-2y +  ... + a_ny = X(x)...(i)
Corresponding homogeneous differential equation is
Dⁿy + a₁D^n-1y + a₂D^n-2y +  ... + a_ny = 0 ...(ii)
then the complementary function is defined as the general solution of the corresponding homogeneous differential equation (ii) and therefore it will invlove n arbitrary constants. The Particular solution is a solution of the complete equation (i) and does not involve any arbitrary constant.

Proof : Substitute
y = e^mx
in the given differential equation



∴ the given differential equation 
a₀m²e^mx + a₁ me^mx + a₂e^mx = 0
⇒ a₀m² + a₁m + a₂ = 0
which is the required auxiliary equation to which the given differential equation reduces to
Remark : Verify that no other substitution will yield the given auxiliary equation.

The given differential equation is
  ...(i)
Its auxiliary equation is given by
D⁴ - 81 = 0

∴ The general solution of equation (i) is given by y =  c₁e^3x + c₂e^-3x + c₃ cos3x + c₄ sin3x.

The auxiliary equation is given by
a₀D² + a₁D + a₂ = 0  .....(i)
Its roots are given by

Clearly,

The constants c₁ and c₂ are arbitrary and are evaluated with the help of prescribed conditions.

The given differential equation is

Its auxiliary equation is D² - 6D + 8 = 0
The roots of the auxiliary equation are m = 2 and 4
Therefore, the general solution of (i) is given by 

Remark : Verify by actual substitution that y = e^x satisfies the given differential equation.

Proof: Let y = c satisfy the given equation

Now

Substituting in (i), we get 
0 - 0 + 12c = 36
⇒ c = 3

The given differential equation is
f(D)y = e^ax   .......(i)
Case I : If f(a) ≠ 0, then a particular solution is given by

[Here D is replaced by a]
Case II : If f(a) = 0, then a particular solution is given by

We illustrate this with the help of following example.
Ex. : Find a particular solution of the differential equation.

and a = 1
so that f(a) =  f(1) = 1 - 4 + 3 = 0
∴ A particular solution is given by


 .....(ii)

Conclusion : It follows that if f(a) = 0, a particular solution will always exist.

If the given differential equation is of the form f(D²)y = sin ax,
so thai f(D²) contains only the even powers of D, then a particular solution is given by

Proof : We have

Remark : Here we used the following general result : If f(a) = 0, then a particular solution of

The given differential equation is

Proof : The given, differential equation is f(D) y = cos 3x,
where f(D) = D² - 2D + 1
Therefore a particular integral is given by

Note that