Which of the following differential equations has $y = {c_1}{e^x} + {c_2}{e^{ - x}}$ as the general solution?
• $\cfrac{{{d^2}y}}{{d{x^2}}} + y = 0$
• $\cfrac{{{d^2}y}}{{d{x^2}}} - y = 0$
• $\cfrac{{{d^2}y}}{{d{x^2}}} + 1 = 0$
• $\cfrac{{{d^2}y}}{{d{x^2}}} - 1 = 0$
.: (B) Given general solution is $y = {c_1}{e^x} + {c_2}{e^{ - x}}$ …(1)
Differentiating (1) w.r.t. $x$,
we get
${y_1} = {c_1}{e^x} + {c_2}{e^{ - x}}( - 1)$
…(2)
Again differentiating (2) w.r.t. $x$,
we get
${y_2} = {c_1}{e^x} - {c_2}{e^{ - x}}( - 1) \Rightarrow {y_2} = {c_1}{e^x} + {c_2}{e^{ - x}}$
$\Rightarrow {y_2} = y$
(using (1))
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Differential Equations. Curated by Sachin Sharma. Free for all students.