Please show work out for number 4 i couldnt solve it. Find a linear homogeneous

Question

Please show work out for number 4 i couldnt solve it.

Find a linear homogeneous constant-coefficient differential equation with the given general solution. y(x) = c1ex + (c2 + c3x + c4x2)e-x Use the Wronskian method to show the following solutions are linearly independent. y1 = x y2 = xex y 3 =1 Given that y1 = et is a solution to the d3y/dt3+5d2y/dt2+5dy/dt-11y = 0 differential equation. Find its general solution. Find the general solution to the following differential equation. D2y/dx-2dy/dx-35y = 1+13sinx-e3x

Explanation / Answer

Let e^(kt) be a solution for the given differential equation.

Substitute this in the equation. We get,

k^3 + 5k^2 + 5K - 11 = 0

Also given that e^t is a solution. So, k = 1 is a solution to the above equation.

To find other solutions, divide the expression ( k^3 + 5k^2 + 5K - 11 ) with (k-1).We get,

(k^2 + 6k + 11). So the other 2 roots are (-3 + i*sqrt(2)) and (-3 - i*sqrt(2)).

We know that e^(ix) = cos(x) + i*sin(x).

Similarly e^((-3 + i*sqrt(2))*i)t = e^(-3t) *(cos sqrt(2)t) + i*e^(-3t) *(sin sqrt(2)t)

e^((-3 - i*sqrt(2))*i)t = e^(-3t) *(cos sqrt(2)t) - i*e^(-3t) *(sin sqrt(2)t)

Therefore the other 2 solutions of given diff. equation are

e^(-3t) *(cos sqrt(2)t) and e^(-3t) *(sin sqrt(2)t)

Finally the general solution will be

a*e^t + b*e^(-3t)*(cos sqrt(2)t) + c*e^(-3t)*(sin sqrt(2)t) , a,b,c are parameters.

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---