Find the differential equation of y = eˣ(a+bx) (a, b constant).

Question 39 of 50

Q. Find the differential equation of y = eˣ(a+bx) (a, b constant).

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2023Subjective· 2mImportance★★★★★

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Differentiate twice to eliminate the two constants $a,b$ ; the derivatives of $y=e^x(a+bx)$ combine neatly because $e^x$ reproduces itself.

Step 1. $y=e^x(a+bx)$ . Differentiate:

$\frac{dy}{dx}=e^x(a+bx)+e^x\cdot b = e^x(a+bx+b) = y+be^x.$

So $\dfrac{dy}{dx}-y = be^x$ . Call this $(*)$ .

Step 2. Differentiate $(*)$ again (i.e. differentiate $y'$ once more):

$\frac{d^2y}{dx^2}=e^x(a+bx+b)+e^x\cdot b = e^x(a+bx+2b) = \frac{dy}{dx}+be^x.$

But from $(*)$ , $be^x=\dfrac{dy}{dx}-y$ , so

$\frac{d^2y}{dx^2}=\frac{dy}{dx}+\left(\frac{dy}{dx}-y\right)=2\frac{dy}{dx}-y.$

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Q.Find the differential equation of y = eˣ(a+bx) (a, b constant).