Notes On Forming a Differential Equation - CBSE Class 12 Maths
Building a differential equation whose general solution is given. xy = ax2 + b/x To find the differential equation, the solution is differentiated to the extent that the constants are eliminated. By multiplying both the sides of the equation by x. x(xy) = x(ax2 + b/x)   ⇒ x2y = ax3 + b x2 dy/dx + 2xy = 3ax2 + 0 ⇒ x dy/dx + 2y = 3ax ...........(I) ⇒ x d2y/dx2 + dy/dx + 2 dy/dx = 3a ⇒ x d2y/dx2 + 3 dy/dx = 3a ⇒ 3a = x d2y/dx2 + 3 dy/dx  .................. (II) From (1) and (2) ⇒ x dy/dx + 2y = (x d2y/dx2 + 3 dy/dx)x ⇒ x dy/dx + 2y = x2 d2y/dx2 + 3x dy/dx ⇒ x2 d2y/dx2 + 3x dy/dx - x dy/dx - 2y = 0 ⇒ x2 d2y/dx2 + 2x dy/dx - 2y = 0 Working rules to form a differential equation: 1. Observe the number of arbitrary constants present in the given general solution. 2. Find the derivative up to the order based on the number of constants. Ex: x2 = 4ay (One constant: a) Differential equation: x. dy/dx = 2y x2/b2 + y2/a2 = 1 (Two constants: a and b) Here the number of arbitrary constants is two, then differentiate the general solution twice.Then we get Differential equation: xy d2y/dx2 + x(dy/dx)2 - y dy/dx = 0 3. Eliminate the arbitrary constants using the equations obtained. xy = cex + be-x + x2 Constants: c,b Differentiating both sides with respect to x: x . dy/dx + y = cex - be-x + 2x……. (1) Differentiating both sides with respect to x again: x d2y/dx2 + dy/dx + dy/dx = cex + be-x + 2 ⇒ x d2y/dx2 + 2. dy/dx = cex + be-x + 2 ….. (2) ⇒ x d2y/dx2 + 2. dy/dx = xy - x2 + 2 [Since, xy = cex + be-x + x2] Required differential equation x d2y/dx2 + 2. dy/dx - xy + x2 - 2 = 0.

#### Summary

Building a differential equation whose general solution is given. xy = ax2 + b/x To find the differential equation, the solution is differentiated to the extent that the constants are eliminated. By multiplying both the sides of the equation by x. x(xy) = x(ax2 + b/x)   ⇒ x2y = ax3 + b x2 dy/dx + 2xy = 3ax2 + 0 ⇒ x dy/dx + 2y = 3ax ...........(I) ⇒ x d2y/dx2 + dy/dx + 2 dy/dx = 3a ⇒ x d2y/dx2 + 3 dy/dx = 3a ⇒ 3a = x d2y/dx2 + 3 dy/dx  .................. (II) From (1) and (2) ⇒ x dy/dx + 2y = (x d2y/dx2 + 3 dy/dx)x ⇒ x dy/dx + 2y = x2 d2y/dx2 + 3x dy/dx ⇒ x2 d2y/dx2 + 3x dy/dx - x dy/dx - 2y = 0 ⇒ x2 d2y/dx2 + 2x dy/dx - 2y = 0 Working rules to form a differential equation: 1. Observe the number of arbitrary constants present in the given general solution. 2. Find the derivative up to the order based on the number of constants. Ex: x2 = 4ay (One constant: a) Differential equation: x. dy/dx = 2y x2/b2 + y2/a2 = 1 (Two constants: a and b) Here the number of arbitrary constants is two, then differentiate the general solution twice.Then we get Differential equation: xy d2y/dx2 + x(dy/dx)2 - y dy/dx = 0 3. Eliminate the arbitrary constants using the equations obtained. xy = cex + be-x + x2 Constants: c,b Differentiating both sides with respect to x: x . dy/dx + y = cex - be-x + 2x……. (1) Differentiating both sides with respect to x again: x d2y/dx2 + dy/dx + dy/dx = cex + be-x + 2 ⇒ x d2y/dx2 + 2. dy/dx = cex + be-x + 2 ….. (2) ⇒ x d2y/dx2 + 2. dy/dx = xy - x2 + 2 [Since, xy = cex + be-x + x2] Required differential equation x d2y/dx2 + 2. dy/dx - xy + x2 - 2 = 0.

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