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Question

Suppose f and g are differentiable functions on (0,) such that f'(x)=g(x)x and g'(x)=f(x)x, for all x>0. Further, f(1)=3 and g(1)=1.

f(10) is equal to

A
10120
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B
10
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C
20
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D
20120
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Solution

The correct option is D 20120
We know that,
f(x)+g(x)=Ax(1)f(x)g(x)=Bx(2)
Adding (1) and (2)
2f(x)=Ax+Bxf(x)=px+qx (Here p=A2, q=B2)f(1)=p+q=3(3)

Subtracting (2) from (1),
2g(x)=AxBxg(x)=pxqxg(1)=pq=1(4)

Using (3) and (4),
2p=2p=1q=2f(x)=1x+2x;g(x)=1x2xf(10)=110+20=20120

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