Solve the differential equation by variation of parameters. y'' + y = sec x

Solution:

Given, the differential equation is y'' + y = sec x.

We have to solve the differential equation by variation of parameters.

The standard form of equation is given by y'' + py’ + qy = X,

Where p, q and X are functions of x.

The complementary function of the equation is given by Yc = C₁Y₁ + C₂Y₂

The complementary function of the given equation is y'' + y = 0

Let y'' = m

Now, m² + 1 = 0

m² = -1

Taking square root,

m = ±i

The solution is of the form α ± βi.

Here, α = 0 and β = 1

So, Yc = C₁.e^αx.cos(βx) + C₂.e^αx.sin(βx)

= C₁cos(1.x) + C₂.sin(1.x)

Yc = C₁ cos(x) + C₂ sin(x)

The particular integral is given by Yp = U₁Y₁ + U₂Y₂

Where, U₁ = ∫W₁/W dx

U₂ = ∫W₂/W dx

W = \(\begin{vmatrix}Y_{1} &Y_{2} \\dY_{1} &dY_{2} \\\end{vmatrix}\)

Here, f(x) = sec x, Y₁ = cos x and Y₂ = sin x

So, dY₁ = - sin x

dY₂’ = cos x

W = \(\begin{vmatrix}cos x &sin x \\-sin x &cos x \\\end{vmatrix}\)

= cosx(cosx) - (-sinx)(sinx)

= cos²x + sin²x

W = 1

W₁ = \(\begin{vmatrix}0 &Y_{2} \\f(x) &dY_{2} \\\end{vmatrix}\)

W₁ = \(\begin{vmatrix}0 &sin x \\sec x &cos x \\\end{vmatrix}\)

= 0(cosx) - sinx(secx)

= -sinx(secx)

= -sinx/cosx

W₁ = -tanx

W₂ = \(\begin{vmatrix}Y_{1} &0 \\dY_{1} &f(x) \\\end{vmatrix}\)

= \(\begin{vmatrix}cosx &0 \\-sinx &sec x \\\end{vmatrix}\)

= cosx(secx) - 0(-sinx)

= cosx/cosx

W₂ = 1

\(U_{1}=\int \frac{-tanx}{1}dx\)

\(U_{1}=\int -tanx\: dx \)

\(U_{1}=\int \frac{-sinx}{cosx}: dx \)

Let u = cos x

du = -sinx dx

Now, \(U_{1}=\int \frac{du}{u}\)

U₁ = ln|cosx|

\(U_{2}=\int \frac{1}{1}\, dx\)

\(U_{2}=\int dx\)

U₂ = x

Now, Yp = ln|cosx|.cosx + x.sinx

We know, the complete solution is Y = Yc + Yp

Y = C₁ cos(x) + C₂ sin(x) + ln|cosx|.cosx + x.sinx

Y = C₁ cos(x) + C₂ sin(x) + cosx ln|cosx| + x sinx

Therefore, the required solution is Y = C₁ cos(x) + C₂ sin(x) + cosx ln|cosx| + x sinx.

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Solve the differential equation by variation of parameters. y'' + y = sec x

Summary:

The solution of the differential equation y'' + y = sec x by variation of parameters is Y = C₁ cos(x) + C₂ sin(x) + cosx ln|cosx| + x sinx.