AP EAMCET · Maths · Differential Equations
The differential equation having \(y=(a+b) e^{c x+d}\) as its general solution, where a, b, c, d, are arbitrary constants, is
- A \(y^{(4)}+3 y y^{(3)}+6 y^{(2)} y^2+y=0\)
- B \(\mathrm{y}^{(3)}+4 \mathrm{yy} \mathrm{y}^{(2)}+6 \mathrm{y}^2 \mathrm{y}^{(1)}+12 \mathrm{y}=0\)
- C \(\mathrm{y}^{(1)}-\mathrm{y}=0\)
- D \(\mathrm{yy}^{(2)}-\left(\mathrm{y}^{(1)}\right)^2=0\)
Answer & Solution
Correct Answer
(D) \(\mathrm{yy}^{(2)}-\left(\mathrm{y}^{(1)}\right)^2=0\)
Step-by-step Solution
Detailed explanation
Given \(y=(a+b) e^{c x+d}\) \(y^1=(a+b) e^{(x+d)}\) \(\begin{aligned} & y^1=c y ; \quad y^2=c y^1 \\ & y^2=\frac{y^{\prime}}{y} \cdot y^1 \\ & \Rightarrow y y^{(2)}-\left(y^{(1)}\right)^2=0\end{aligned}\)
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