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  3. Form the differential equation representing the family of curves , where ‘a’ and ‘b’ are arbitrary constants.
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Form the differential equation representing the family of curves y=e^{2x}(a+bx), where ‘a’ and ‘b’ are arbitrary constants. 

 

 

 

 
 
 
 
 

Answers (1)
R Ravindra Pindel

y=e^{2x}(a+bx)  [given]       ---(1)

Differentiating with respect to x, we get

\frac{\mathrm{d}y }{\mathrm{d} x}=e^{2x}(b)+2(a+bx)e^{2x}

\frac{\mathrm{d}y }{\mathrm{d} x}=be^{2x}+2y           --from (1)

\frac{\mathrm{d}y }{\mathrm{d} x}-2y=be^{2x}           --(2)

Again, differentiating with respect to x, we get

\frac{\mathrm{d^2}y }{\mathrm{d} x^2}-2\frac{\mathrm{d} y}{\mathrm{d} x}=2be^{2x}

\frac{\mathrm{d^2}y }{\mathrm{d} x^2}-2\frac{\mathrm{d} y}{\mathrm{d} x}=2\left ( \frac{\mathrm{d} y}{\mathrm{d} x}-2y \right )          -- from (2)

\frac{\mathrm{d^2}y }{\mathrm{d} x^2}-2\frac{\mathrm{d} y}{\mathrm{d} x}=2\frac{\mathrm{d} y}{\mathrm{d} x}-4y

\frac{\mathrm{d^2}y }{\mathrm{d} x^2}-4\frac{\mathrm{d} y}{\mathrm{d} x}+4y=0

This is the required differential equation.

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