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BITSAT 2020
Exam
BITSAT
Year
2020
Subject
Mathematics
Chapter
Differential Equations

BITSAT 2020 · Mathematics · Differential Equations

The solution of the differential equation (x+1) (d y)/(d x)-y=e^ 3...

The solution of the differential equation (x+1)dydx−y=e3x(x+1)2(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^{2} is

Options

  1. Ay=(x+1)e3x+cy=(x+1) e^{3 x}+c
  2. B3y=(x+1)+e3x+c3 y=(x+1)+e^{3 x}+c
  3. C3yx+1=e3x+c\frac{3 y}{x+1}=e^{3 x}+c
  4. Dye−3x=3(x+1)+cy e^{-3 x}=3(x+1)+c

Correct answer

C. 3yx+1=e3x+c\frac{3 y}{x+1}=e^{3 x}+c

Step-by-step solution

The given equation is dydx−yx+1=e3x(x+1)\frac{d y}{d x}-\frac{y}{x+1}=e^{3 x}(x+1) I.F. =e∫−1x+1dx=e−log⁡(x+1)=1x+1=e^{\int-\frac{1}{x+1} d x}=e^{-\log (x+1)}=\frac{1}{x+1} The solution is y(1x+1)=∫e3x(x+1)⋅1x+1dx+a⇒yx+1=∫e3xdx+a=e3x3+a⇒3yx+1=e3x+c,c=3ay\left(\frac{1}{x+1}\right)=\int e^{3 x}(x+1) \cdot \frac{1}{x+1} d x+a\Rightarrow \frac{y}{x+1}=\int e^{3 x} d x+a=\frac{e^{3 x}}{3}+a\Rightarrow \frac{3 y}{x+1}=e^{3 x}+c, c=3 a
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