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Show that each of the following expressions is a solution of the corresponding given differential equation.(i) y = 2x2; xy’ = 2y(ii) y = aex + be-x; y” – y = 0 |
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Answer» (i) v = 2x2 … (1) Differential equation: xy’ = 2y Differentiate with respect to ‘x’ y' = 4x x = y'/4 Substitute in (1) y = 2x(x) y = 2x(y'/4) = xy'/2 On simplifying, 2y = xy’ ∴ (1) is solution of the given differential equation. (ii) y = aex + be-x …(1) Differential equation: y” – y = 0 Differentiate with respect to ‘x’ y’ = aex – be-x Again differentiate with respect to ‘x’ y” = aex + be-x y” = y ⇒ y” – y = 0 ∴ (1) is the solution of the given differential equation. |
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