AP EAMCET Engineering 20 Apr 2019 Shift 1 Paper

Simplify the given matrix.
|

f(x)	f′(x)
f′(x)	f′′(x)

|‌‌=0
f(x)f′′(x)−(f′(x))2‌‌=0
f(x)f′′(x)‌‌=(f′(x))2
‌

f′′(x)

f′(x)

‌‌=‌

f′(x)

f(x)

Integrate both the sides.
∫‌

f′′(x)

f′(x)

dx‌‌=∫‌

f′(x)

f(x)

dx
loge(f′(x))‌‌=loge(f(x))+c‌‌......(1)
since f(0)=1 and f′(0)=2 then
c=loge2
Substituting the value of c in equation (1), we get
loge(f′(x))‌‌=loge(f(x))+loge2
f′(x)‌‌=2f(x)
‌

f′(x)

f(x)

‌‌=2
Integrating both sides w. r. t. x
∫‌

f′(x)

f(x)

dx=∫2dx
logef(x)=2x+c
For f(0)=1,c=0
This gives,
logef(x)‌‌=2x
f(x)‌‌=e2x