Question

Match the following differential equations with their solutions. The symbols A, B, C in the solutions stand for arbitrary constants. You must get all of the answers correct to receive credit.
1. d^2y/dx^2 + 25y = 0
2. dy/dx = 2xy/x^2 - 5y^2
3. d62y/dx^2 + 16 dy/dx + 64y = 0
4. dy/dx = 10xy
5. dy/dx + 24x^2y = 24 x^2
A. y = Ce^-8x^3 + 1
B. 3yx^2 - 5y^3 = C
C. y = Ae^-8x + Bxe^-8x
D. y = Ae^5x^2
E. y = A cos(5x) + B sin(5x)

Answer 1

Answer:

$1 \rightarrow E, 2\rightarrow B, 3\rightarrow C, 4\rightarrow D, 5\rightarrow A$

Step-by-step explanation:

1.     $\frac{d^2y}{dx^2}+25y=0$

    The characteristic equation for the given differential equation is:

        $r^{2} +25=0$

    $\Rightarrow r^2=-25$

    $\Rightarrow r=\pm 5i$

   Since the roots are complex

Now, the general solution is:

 $y=A\cos 5x+B\sin 5x$

2.    $\frac{dy}{dx}=\frac {2xy}{x^2}-5y^2$

     $\Rightarrow \frac{dy}{dx}-\frac 2xy=-5y^2$

  Divide both sides by $y^{-1}$

  Let, $v=y^{-1} \Rightarrow \frac{dv}{dx}=-y^{-2}\frac{dy}{dx}$

    $\Rightarrow -\frac{dv}{dx}-\frac 2xv=-5$

    $\Rightarrow \frac{dv}{dx}+\frac 2xv=5$

 Here, $p(x)=\frac 2x\; \text{and}\;\; q(x)=5$

 I.F. $=e^{\int \frac 2xdx}=x^2$

 Now, the general solution is:

    $vx^2=\int x^2 5dx=\frac {5x^3}3+c$

      $\Rightarrow \frac {x^2}y-\frac {5x^3}3=c$

      $\Rightarrow 3x^2-5x^3y=Cy$

3.     $\frac{d^2y}{dx^2}+16\frac{dy}{dx}+64y=0$

   The characteristic equation is:

     $r^2+16r+64=0$

  $\Rightarrow r^2+8r+8r+64=0$

  $\Rightarrow r(r+8)+8(r+8)=0$

  $\Rightarrow (r+8)(r+8)=0$

  $\Rightarrow r=-8,-8$

 Since the roots are real and repeated.

 Now, the general solution is:

 $y=Ae^{-8x}+Bxe^{-8x}$

4.    $\frac {dy}{dx}=10xy$

   $\Rightarrow \frac {dy}{y}=10xdx$

 Integrating both sides

     $\int\frac {dy}y=\int 10xdx+\log c$

   $\Rightarrow \log y=5x^2+\log c$

   $\Rightarrow y=e^{5x^2}+c$

5.      $\frac {dy}{dx}+24x^2y=24x^2$

      Here, $p(x)=24x^2 \; \text{and}\;\; q(x)=24x^2$

  I.F.$= e^{\int 24x^2dx}=e^{8x^3}$

  Now, the general solution is:

     $y.e^{8x^3}=\int 24x^2 e^{8x^3}dx=24\int x^2e^{8x^3}dx$

               Let, $8x^3=t \Rightarrow 24x^2dx=dt\Rightarrow x^2dx=\frac {dt}{24}$

   $\Rightarrow ye^{8x^3}=\int e^tdt$

   $\Rightarrow ye^{8x^3}=e^{8x^3}+c$

   $\Rightarrow y=1+ce^{-8x^3}$