Chapter 8: Q17P (page 436)

Solve the following equations using method (d) above. $x^{2} y^{''} + x y^{'} - 16 y = 8 x^{4}$

Short Answer

Expert verified

The general solution of the equation is $y = c_{1} x^{- 4} + c_{2} x^{4} + (l n x) x^{4}$

Step by step solution

Given information

The given differential equation is $x^{2} y^{''} + 3 x y^{'} - 3 y = 8 x^{4}$ .

Auxiliary equation

Auxiliary equation is an algebraic equation of degree $n$ upon which depends the solution of a given $n^{t h}$ -order differential equation or difference equation.

Solve for double differential

Consider the given equation.

$x^{2} y^{''} + 3 x y^{'} - 3 y = 0 Let x = e^{z} . So, x = e^{z} z = l n x$

$\frac{d z}{d x} = \frac{1}{x}$

Now,

And,

$x \frac{d y}{d x} = \frac{d y}{d z}$

Roots of equation

The given differential equation can be written as,

$(D (D - 1) + D - 16) y = 8 e^{4 z}$

The auxiliary equation of the above equation is,

$m (m - 1) + m - 16 = 0$

The solution of the auxiliary equation is,

$m = \pm 4$

Let the roots be represented as,

$a = - 4 b = 4$

Solve for Yp

Now,

$Q = 8 e^{4 z}$

So,

Now,

$D$ represent the first derivative and $D^{2}$ represent the second derivative.

So,

Complete solution

Thus, the complete solution is given as,

Therefore, the general solution of the equation is $y = c_{1} x^{- 4} + c_{2} x^{4} + (l n x) x^{4}$

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Solve the following equations using method (d) above. $x^{2} y^{''} + x y^{'} - 16 y = 8 x^{4}$

Short Answer

Step by step solution

Given information

Auxiliary equation

Solve for double differential

Roots of equation

Solve for Yp

Complete solution

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Solve the following equations using method (d) above. x2y''+xy'-16y=8x4

Short Answer

Step by step solution

Given information

Auxiliary equation

Solve for double differential

Roots of equation

Solve for Yp

Complete solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Force

Space Physics

Physical Quantities And Units

Wave Optics

Conservation of Energy and Momentum

Medical Physics

Study anywhere. Anytime. Across all devices.

Solve the following equations using method (d) above. $x^{2} y^{''} + x y^{'} - 16 y = 8 x^{4}$