If a curve y = f(x) passes through the point (1, 2) and satisfies [x dy dx + y = bx^4], th…

Question

If a curve y = f(x) passes through the point (1, 2) and satisfies $$x {{dy} \over {dx}} + y = b{x^4}$$, then for what value of b, $$\int\limits_1^2 {f(x)dx = {{62} \over 5}} $$?

Accepted Answer

$${{dy} \over {dx}} + {y \over x} = b{x^3}$$, $$I.F. = {e^{\int {{{dx} \over x}} }} = x$$

$$ \therefore $$ $$yx = \int {b{x^4}dx}  = {{b{x^5}} \over 5} + C$$

Passes through (1, 2), we get

$$2 = {b \over 5} + C$$ ........ (i)

Also, $$\int\limits_1^2 {\left( {{{b{x^4}} \over 5} + {c \over x}} \right)dx = {{62} \over 5}} $$

$$ \Rightarrow {b \over {25}} \times 32 + C\ln 2 - {b \over {25}} = {{62} \over 5} \Rightarrow C = 0$$ \& $b = 10$

Solutions · 1