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Q16E

Expert-verifiedFound in: Page 453

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**In Problems 15-17,solve the given initial value problem x ^{2}y"+5xy'+4y=0.**

**y(1) =3 and y'(1) = 7**

The solution of the given initial value problem is y=3x^{-2}+13x^{-2} ln x.

**In mathematics, a Cauchy problem is one in which the solution to a partial differential equation must satisfy specific constraints specified on a hypersurface in the domain. **

**An initial value problem or a boundary value problem is an example of the Cauchy problem. **

**The equation will be in the form of, ax ^{2}y"+bxy'+cy=0.**

The given equation is,

x^{2}y"+5xy'+4y=0

Let L be the differential operator defined by the left-hand side of equation, that is,

L [x] (t) = x^{2}y"+5xy'+4y

And let.

w (r,t) = x^{r}

Substituting the w(r,t) in place of x(t), you get

L [w] (t)=x^{2} (x^{r})"+5x (x^{r})'+4 (x^{r})

=x^{2 }(r (r-1)) x^{r-2 }+5x (r) x^{r-1} +4x^{r}

= (r^{2}-r) x^{r} +5rx^{r} +4x^{r}

=(r^{2}+4r+4) x^{r}

Solving the indicial equation,

r^{2}+4r+4 =0

(r+2)^{2} =0

There are repeated roots at r= -2

Thus there are two linearly independent solutions given by

y_{1}=c_{1}x^{-2} and y_{2} = c_{2}x^{-2} lnx

The general solution for the equation will be

y=c_{1}x^{-2} +c_{2}x^{-2} lnx

For the given initial conditions,

y(1)=3 and y'(1)=7

y(x)=c_{1} x^{-2}+c_{2} x^{-2} lnx

y(1) = c_{1}

c_{1}=3

y'(x)=c_{1} (-2) x^{-3} +c_{2} [(-2) x^{-3} lnx+x^{-2} (1/x)]

y'(1) = -2c_{1}+c_{2}

-2c_{1}+c_{2}=7

Solving the two simultaneous equations (1) and (2), you get the values of two constants c_{1} and c_{2} as,

c_{1} = 3 and c_{2} = 13

Thus the solution of the given initial value problem is y=13 x^{-2}+13x^{-2} ln x.

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