AMU MCA Differential Equation Previous Year Questions (PYQs) – Page 1 of 3
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The solution of the linear difference equation $y_{k+1}-ay_k=0$, $a\ne 1$ is
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Solution
$y_{k+1}=ay_k \Rightarrow y_k=ca^k$
The general solution of the differential equation
$(D^2 - a^2)^3 y = e^{ax}$, where $D=\frac{d}{dx}$ is
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Solution
Characteristic equation:
$(m^2-a^2)^3=0$
Roots: $m=\pm a$ each of multiplicity 3.
Hence complementary function:
$(c_1+c_2x+c_3x^2)e^{ax}+(c_4+c_5x+c_6x^2)e^{-ax}$
Since RHS is $e^{ax}$ and $m=a$ has multiplicity 3,
Particular integral = $x^3\frac{e^{ax}}{3!\,(2a)^3} =\frac{x^2e^{ax}}{8a^2}$
If $a$ is an arbitrary constant, then solution of the differential equation $\dfrac{dy}{dx} + \sqrt{\dfrac{1-y^2}{1-x^2}}=0$ is
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Solution
The solution of the equation
$ x^2 \dfrac{d^2 y}{dx^2} = \log x $
when $ x = 1,; y = 0 $ and $ \dfrac{dy}{dx} = -1 $ is
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Solution
Solution of differential equation
$y^2\log y = xy\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^2$ is
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Solution
The solution of the differential equation
$\frac{dy}{dx}=\frac{2xy-y^2}{2xy-x^2}$ is
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Solution
$y=a\cos(\log x)+b\sin(\log x)$ is a solution of the differential equation
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Solution
The integrating factor of the D.E.
$(x\log x)\frac{dy}{dx}+y=2\log x$ is
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Solution
Using section formula,
$ \vec c=\frac{m\vec b+1\cdot \vec a}{m+1} $
Comparing with $\vec c=\lambda \vec a+\mu \vec b$,
$ \lambda=\frac{1}{m+1},\ \mu=\frac{m}{m+1} $
$ \Rightarrow \lambda+\mu=\frac{1+m}{m+1}=1 $
The general solution of differential equation
$\frac{dy}{dx}+2xy=2e^{-x^2}$ is
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Solution
$ \frac{dy}{dx}+2xy=2e^{-x^2} $
Here, $P=2x$
$ \text{I.F.}=e^{\int 2x dx}=e^{x^2} $
Multiply both sides:
$ y e^{x^2}=\int 2e^{-x^2} \cdot e^{x^2} dx + C $
$ =\int 2dx + C =2x + C $
$ \Rightarrow y=(2x+C)e^{-x^2} $
The Laplace transform of $\sin \sqrt{x}$ is:
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Solution
The feasible region allows $z$ to increase indefinitely, hence the objective function is unbounded.
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