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Ex 9.6
Ex 9.6, 2
Ex 9.6, 3 Important
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Ex 9.6, 5 Important
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Ex 9.6, 7 Important
Ex 9.6, 8 Important
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Ex 9.6, 12 Important
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Ex 9.6, 15
Ex 9.6, 16 Important You are here
Ex 9.6, 17 Important
Ex 9.6, 18 (MCQ)
Ex 9.6, 19 (MCQ) Important
Last updated at Jan. 3, 2020 by Teachoo
Ex 9.6, 16 Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (π₯ , π¦) is equal to the sum of the coordinates of the point. We know that Slope of tangent to curve at (x, y) = ππ¦/ππ₯ Given that Slope of the tangent to the curve at any point (π₯ , π¦) is equal to the sum of the coordinates of the point. Therefore, ππ¦/ππ₯ = x + y ππ¦/ππ₯ β y = x This is the form ππ¦/ππ₯+Py=Q where P = β1 & Q = x Finding Integrating factor IF = π^β«1βγπππ₯ γ IF = π^β«1βγ(β1)ππ₯ γ IF = eβx Solution is y(IF) = β«1βγ(πΓπΌπΉ)ππ₯+πγ π¦π^(βπ₯) = β«1βγπ₯π^(βπ₯) ππ₯+πγ yeβx = x β«1βγπ^(βπ₯) ππ₯βγ β«1βγ[1β«1βγπ^(βπ₯) ππ₯γ] ππ₯+πγ yeβx = βx π^(βπ₯) ββ«1βγβπ^(βπ₯) ππ₯+πγ yeβx = βx π^(βπ₯)+β«1βγπ^(βπ₯) ππ₯+πγ yeβx = βx π^(βπ₯)+(βπ^(βπ₯))/(β1) +π yeβx = βx π^(βπ₯)βπ^(βπ₯) +π Dividing both sides by eβx y = βx β 1 + π/π^(βπ₯) Integrating by parts with β«1βγπ(π₯) π(π₯) ππ₯=π(π₯) β«1βγπ(π₯) ππ₯ ββ«1βγ[π^β² (π₯) β«1βγπ(π₯) ππ₯] ππ₯γγγγ Take f (x) = x & g (x) = π^(βπ₯) y = βx β 1 + cex Since curve passes through origin, Putting x = 0 & y = 0 in (2) 0 = 0 β 1 + Ce0 1 = C C = 1 Put value of C in (1) y = βx β 1 + ex x + y + 1 = ex β¦(1)