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Chapter 1: Q5E (page 1)

Find a general solution for the differential equation with x as the independent variable:

ym+3yn+28y'+26y=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is y=c1e鈭�x+c2e鈭�xcos5x+c3e鈭�xsin5x

Step by step solution

01

Auxiliary equation:

The auxiliary equation for y'''+3y''+28y'+26=0is r3+3r2+28r+26=0. The solution of previous equation is:

r3+3r2+28r+26=0(r+1)(r2+2r+26)=0r=鈭�1r2+2r+26=0r2,3=鈭�2卤22鈭�4.262=鈭�1卤5i

Where I is imaginary unit

02

General solution:

Then 伪=鈭�1,尾=5 and we can write the general solution for the differential equation:

y=c1e鈭�x+c2e鈭�xcos5x+c3e鈭�xsin5x

Hence the final solution is y=c1e鈭�x+c2e鈭�xcos5x+c3e鈭�xsin5x

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Most popular questions from this chapter

Question: Let f(x)and g(x)be analytic at x0. Determine whether the following statements are always true or sometimes false:

(a) 3f(x)+g(x)is analytic at x0.

(b) f(x)/g(x)is analytic at x0.

(c) f'(x)is analytic at x0.

(d) f(x)3-鈭�x0xg(t)dt is analytic at x0.

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

x2-sin(x+y)=1,dydx=2xsec(x+y)-1

Verify that 蠒(x)=2(1-cex),where c is an arbitrary constant, it is a one-parameter family of solutions to dydx=y(y-2)2. Graph the solution curves corresponding to c=0,卤1,卤2 using the same coordinate axes.

Question: Show that,28.鈭�n=0鈭�anxn+1+鈭�n=1鈭�nbnxn-1=b1+鈭�n-1鈭�[2an-1+(n+1)bn+1]xn

In Problems 3-8, determine whether the given function is a solution to the given differential equation.

y=sinx+x2,d2ydx2+y=x2+2
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