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Messages - Tiantian Yu

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Quiz-4 / TUT0303 Quiz4

« on: October 18, 2019, 02:00:00 PM »

Find the general solution of the given differential equation: y''+4y'+6.25y=0

solution: The auxiliary equation is \begin{equation*}r^2+4r+6.25=0\end{equation*}
By the quadratic formula, the roots are \begin{equation*}r=\frac{-4\pm\sqrt{4^2 - 4(6.25)}}{2}=-2\pm\frac{3}{2}i.\end{equation*}
If the roots of the auxiliary equation are the complex numbers 􏲬\begin{equation*}r=\alpha\pm i\beta\end{equation*} then the general solution of ay''+by'+cy=0 is\begin{equation*} y=e^{\alpha t} (C_1 cos\beta t +C_2sin\beta t)\end{equation*}
since􏲬\begin{equation*}\alpha=-2,\beta=\frac{3}{2}\end{equation*}
the general solution of the differential equation is \begin{equation*} y=e^{-2 t} (C_1 cos\frac{3}{2} t +C_2sin\frac{3}{2} t)\end{equation*}

Quiz-3 / TUT0303 Quiz3

« on: October 11, 2019, 02:00:39 PM »

Verify that the functions y₁ and y₂ are solutions of the given differential equation. Do they constitute a fundamental set of solutions?
y'' + 4y = 0; y₁(t)=cos(2t), y₂(t)=sin(2t)

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solution: firstly verifying that the functions y₁ and y₂ are solutions of the given differential equation
verify y₁(t)=cos(2t):
Left-hand side=y'' + 4y=(cos(2t))''+4(cos(2t))
=(-2sin(2t))'+4(cos(2t))
=-4cos(2t)+4(cos(2t))
=0=Right-hand side

verify y₂(t)=sin(2t):
Left-hand side=y'' + 4y=((sin(2t))''+4(sin(2t))
=((2cos(2t))'+4(sin(2t))
=(-4sin(2t)+4(sin(2t))
=0=Right-hand side
Thus the functions y₁ and y₂ are solutions of the given differential equation.

Now we are checking if the given functions constitute a fundamental set of solutions
We calculate the Wronskian of y₁ and y₂:
\begin{equation*}
W=\begin{bmatrix}
cos(2t) & sin(2t)\\
-2sin(2t) & 2cos(2t)
\end{bmatrix}\end{equation*}
\begin{equation*}=cos(2t)(2cos(2t))-[sin(2t)(-2sin(2t)]\end{equation*}
\begin{equation*}=2cos^2(2t)-(-2sin^2(2t))\end{equation*}
\begin{equation*}=2(cos^2(2t)+sin^2(2t))\end{equation*}
\begin{equation*}=2\end{equation*}
Since the Wronskian of y₁ and y₂ is not zero, which means they are linearly independent, the functions y₁ and y₂ constitute a fundamental set of solutions

Quiz-2 / TUT0303 Quiz2

« on: October 04, 2019, 05:57:49 PM »

find the value of b for which the given equation is exact, and then solve it using that value of b.
\begin{equation*}
(xy^2+bx^2y)+(x+y)x^2y'=0
\end{equation*}
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Solution: By calculating M_y and N_x , we find that
\begin{equation*} M_y(x,y)= \frac{d}{dy} (xy^2+bx^2y) = 2xy+bx^2 \end{equation*} \begin{equation*} N_x(x,y)= \frac{d}{dx} ((x+y)x^2) = \frac{d}{dx} (x^3+yx^2)= 3x^2+2xy \end{equation*}
since we know that the given equation is exact, and it is exact when M_y=N_x
\begin{equation*}2xy+bx^2=3x^2+2xy\end{equation*}
thus we get b=3

there is a ψ(x, y) such that
\begin{equation*}ψ_x(x,y)=xy^2+3x^2y \end{equation*} \begin{equation*}ψ_y(x,y)=(x+y)x^2=x^3+yx^2\end{equation*}
Integrating the first of these equations with respect to x, we obtain \begin{equation*}ψ (x , y )=\int(xy^2+3x^2y) dx=\frac{1}{2} x^2y^2+x^3y+h(y) \end{equation*}
Then computing ψ_y from the equation above and setting ψ_y = N gives
\begin{equation*}ψ_y(x,y) =\frac{d}{dy} (\frac{1}{2} x^2y^2+x^3y+h(y))=x^2y+x^3+h'(y)=x^3+yx^2\end{equation*}
Thus h′(y)=0, and h(y)=C
Substituting for h(y) in ψ(x, y) gives \begin{equation*}ψ(x,y) =\frac{1}{2} x^2y^2+x^3y+C \end{equation*}
Hence the solution is given implicitly by \begin{equation*}C=\frac{1}{2} x^2y^2+x^3y=x^2y^2+2x^3y\end{equation*}