CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Differential Equation

Q.2. Form the differential equation of the family of circles having centre on x-axis and passing through the origin.

Solution :

Let the radius of the circle be a.
Therefore, Centre = (a, 0).
The equation of circle is (x – a)² + (y – 0)² = a² [Using : (x – h)² + (y – k)² = r²]
Or, x² + a² – 2ax + y² – a2 = 0
Or, x² + y² – 2ax = 0 ----------------------- (1)
Differentiating w.r.t. x, we get
2x + 2ydy/dx – 2a = 0
Or, a = x + ydy/dx
Putting the value of a in (1), we get
x² + y² – 2(x + ydy/dx).x = 0
Or, x² + y² – 2x² – 2xydy/dx = 0
Or, y² – x² – 2xydy/dx = 0
Or, 2xydy/dx + x² – y² = 0 [Ans.]

Q.3. Verify that y = 3 cos (log x) + 4 sin (log x) is a solution of the differential equation,

x2 d²y/dx² + x dy/dx + y = 0.

Solution :

We have, y = 3 cos (log x) + 4 sin (log x)
Differentiating w.r.t., x we get
dy/dx = – 3 sin (log x).1/x + 4 cos (log x).1/x
Therefore, xdy/dx = – 3 sin (log x) + 4 cos (log x) --------------------------(1)
Differentiating (1) w.r.t., x we get
xd²y/dx² + dy/dx.1 = – 3cos (log x).1/x – 4 sin (log x).1/x
Or, xd²y/dx² + dy/dx = – [3 cos (log x) + 4 sin (log x)]/x
Or, x² d²y/dx² + x dy/dx = – y
Or, x² d²y/dx² + x dy/dx + y = 0
Hence, y = 3 cos (log x) + 4 sin (log x) is a solution of x² d²y/dx² + x dy/dx + y = 0.
[Proved.]

9.4. Differential Equations with Variables Separable.

Q.1. Solve the following differential equation :

dy/dx = log (x + 1).

Solution :

We have, dy/dx = log (x + 1) --------------- (1)
Seperating the variables we get,
dy = log (x + 1) dx
Integrating both sides, we get
∫dy = ∫1.log (x + 1) dx + c
Or, y = x log(x + 1) – ∫x.[1/(x + 1)]dx + c [Integrating by parts]
= x log(x +1) – ∫[{(x + 1) – 1}/(x + 1)]dx + c
= x log (x + 1) – ∫[1 – 1/(x + 1)]dx + c
= x log (x + 1) – x + log (x + 1) + c
= (x + 1) log (x + 1) – x + c. [Ans.]

Paper By Mr. M.P.Keshari
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Ph No. : 09434150289