Solve the initial value problem:
𝑥
𝑑
𝑦
𝑑
𝑥
−
3
𝑦
=
𝑥
4
cos
⁡
𝑥
,
𝑦
(
2
𝜋
)
=
0.

Hint: Use an integrating factor.

Question

Solve the initial value problem:   
𝑥
𝑑
𝑦
𝑑
𝑥
−
3
𝑦
=
𝑥
4
cos
⁡
𝑥
,
𝑦
(
2
𝜋
)
=
0.

Hint: Use an integrating factor.

BlackTom AI · Accepted Answer

The problem asks us to solve the initial value problem: x dy/dx − 3y = x^4 cos x, with y(2π) = 0, and hints to use an integrating factor.

First, rewrite the equation in standard linear form for y(x): dy/dx − (3/x) y = x^3 cos x, valid for x ≠ 0. This makes it clear we’re dealing with a first-order linear ODE.

Next, identify the integrating factor μ(x) = exp(∫ −3/x dx) = exp(−3 ln x) = x^−3. Multiplying every term by μ(x) gives: x^−3 dy/dx − 3 x^−4 y = x^0 cos x, i.e., d/dx [y x^−3] = cos x.

Integrate both sides with respect to x: y x^−3 = ∫ cos x dx = sin x + C, where C is a constant of integration. Therefore y = x^3 (sin x + C) = x^3 sin x + C x^3.

Apply the initial condition y(2π) = 0: 0 = (2π)^3 sin(2π) + C (2π)^3. Since sin(2π) = 0, the equation reduces to 0 = C (2π)^3, which implies C = 0.

Thus the particular solution satisfying the initial condition is y(x) = x^3 sin x.

If needed, you can verify by differentiation: dy/dx = 3x^2 sin x + x^3 cos x, and substituting into x dy/dx − 3y yields x(3x^2 sin x + x^3 cos x) − 3x^3 sin x = x^4 cos x, confirming the ODE is satisfied. The initial condition y(2π) = 0 also holds since sin(2π) = 0.

SMAT011 Week 8 Practice Quiz

Solve the initial value problem: 𝑥 𝑑 𝑦 𝑑 𝑥 − 3 𝑦 = 𝑥 4 cos ⁡ 𝑥 , 𝑦 ( 2 𝜋 ) = 0. Hint: Use an integrating factor.

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