Exercise Set 14

  1. Using the method of separation of variables, find the general solutions of the following first order differential equations and the particular solutions in those cases where initial conditions are specified:

    1. \(\displaystyle x^2\frac{dy}{dx}=(y+xy)\)
    2. \(\displaystyle \frac{dy}{dx}=\frac{y-1}{x-1},\; y(2)=3\)
    3. \(\displaystyle \frac{dy}{dx}=\frac{y+2}{x-2}\)

  2. Using the integrating factor method, find the general solutions of the following linear first order differential equations and the particular solutions in those cases where initial conditions are specified:

    1. \(\displaystyle \frac{dy}{dx}+\frac{3}{x}y=4+x^2,\;y(1)=1\)
    2. \(\displaystyle x\frac{dy}{dx}+2y=x^{-4},\;y(1)=-\frac{1}{2}\)
    3. \(\displaystyle \frac{dy}{dx}+\frac{y}{x}=\cos(x),\;y\left(\frac{\pi}{2}\right)=0\)
    4. \(\displaystyle \frac{dy}{dx}=e^{-x}-2y,\;y(0)=0\)
    5. \(\displaystyle \frac{dy}{dx}=y\tan(x)+\sin(x)\)

  3. Find the general solutions of the following constant coefficient, linear, homogeneous differential equations, and the particular solutions in those cases where initial conditions are specified:

    1. \(\displaystyle \frac{d^2y}{dx^2}+5\frac{dy}{dx}+6y=0,\;y(0)=0,\; \frac{dy(0)}{dx}=1\)
    2. \(\displaystyle \frac{d^2y}{dx^2}+8\frac{dy}{dx}+25y=0,\;y(0)=1,\; \frac{dy(0)}{dx}=0\)
    3. \(\displaystyle \frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=0\)

  4. For each of the following differential equations, find: the Complementary Function; a Particular Integral; and the Particular Solution corresponding to the given initial conditions:

    1. \(\displaystyle \frac{d^2y}{dx^2}+\frac{dy}{dx}+y=36e^{5x}\)
    2. \(\displaystyle \frac{d^2y}{dx^2}+3\frac{dy}{dx}-4y=-34\sin(x)\)
    3. \(\displaystyle \frac{d^2y}{dx^2}+3\frac{dy}{dx}-4y=24e^{-x},\;y(0)=0,\; \frac{dy(0)}{dx}=10\)
    4. \(\displaystyle \frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=x^2-2x+3,\;y(0)=\frac{3}{4},\; \frac{dy(0)}{dx}=\frac{3}{2}.\)