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# 5.6E: Reduction of Order (Exercises)

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In Exercises [exer:5.6.1} –[exer:5.6.17} find the general solution, given that $$y_1$$ satisfies the complementary equation. As a byproduct, find a fundamental set of solutions of the complementary equation.

[exer:5.6.1] $$(2x+1)y''-2y'-(2x+3)y=(2x+1)^2; \quad y_1=e^{-x}$$

[exer:5.6.2] $$x^2y''+xy'-y={4\over x^2}; \quad y_1=x$$

[exer:5.6.3] $$x^2y''-xy'+y=x; \quad y_1=x$$

[exer:5.6.4] $$y''-3y'+2y={1\over1+e^{-x}}; \quad y_1=e^{2x}$$

[exer:5.6.5] $$y''-2y'+y=7x^{3/2}e^x; \quad y_1=e^x$$

[exer:5.6.6] $$4x^2y''+(4x-8x^2)y'+(4x^2-4x-1)y=4x^{1/2}e^x(1+4x); \quad y_1=x^{1/2}e^x$$

[exer:5.6.7] $$y''-2y'+2y=e^x\sec x; \quad y_1=e^x\cos x$$

[exer:5.6.8] $$y''+4xy'+(4x^2+2)y=8e^{-x(x+2)}; \quad y_1=e^{-x^2}$$

[exer:5.6.9] $$x^2y''+xy'-4y=-6x-4; \quad y_1=x^2$$

[exer:5.6.10] $$x^2y''+2x(x-1)y'+(x^2-2x+2)y=x^3e^{2x}; \quad y_1=xe^{-x}$$

[exer:5.6.11] $$x^2y''-x(2x-1)y'+(x^2-x-1)y=x^2e^x; \quad y_1=xe^x$$

[exer:5.6.12] $$(1-2x)y''+2y'+(2x-3)y=(1-4x+4x^2)e^x; \quad y_1=e^x$$

[exer:5.6.13] $$x^2y''-3xy'+4y=4x^4; \quad y_1=x^2$$

[exer:5.6.14] $$2xy''+(4x+1)y'+(2x+1)y=3x^{1/2}e^{-x}; \quad y_1=e^{-x}$$

[exer:5.6.15] $$xy''-(2x+1)y'+(x+1)y=-e^x; \quad y_1=e^x$$

[exer:5.6.16] $$4x^2y''-4x(x+1)y'+(2x+3)y=4x^{5/2}e^{2x}; \quad y_1=x^{1/2}$$

[exer:5.6.17] $$x^2y''-5xy'+8y=4x^2; \quad y_1=x^2$$

[exer:5.6.18] $$xy''+(2-2x)y'+(x-2)y=0; \quad y_1=e^x$$

[exer:5.6.19] $$x^2y''-4xy'+6y=0; \quad y_1=x^2$$

[exer:5.6.20] $$x^2(\ln |x|)^2y''-(2x \ln |x|)y'+(2+\ln |x|)y=0; \quad y_1=\ln |x|$$

[exer:5.6.21] $$4xy''+2y'+y=0; \quad y_1=\sin \sqrt{x}$$

[exer:5.6.22] $$xy''-(2x+2)y'+(x+2)y=0; \quad y_1=e^x$$

[exer:5.6.23] $$x^2y''-(2a-1)xy'+a^2y=0; \quad y_1=x^a$$

[exer:5.6.24] $$x^2y''-2xy'+(x^2+2)y=0; \quad y_1=x \sin x$$

[exer:5.6.25] $$xy''-(4x+1)y'+(4x+2)y=0; \quad y_1=e^{2x}$$

[exer:5.6.26] $$4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0; \quad y_1=x^{1/2}$$

[exer:5.6.27] $$4x^2y''-4xy'+(3-16x^2)y=0; \quad y_1=x^{1/2}e^{2x}$$

[exer:5.6.28] $$(2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0; \quad y_1=1/x$$

[exer:5.6.29] $$(x^2-2x)y''+(2-x^2)y'+(2x-2)y=0; \quad y_1=e^x$$

[exer:5.6.30] $$xy''-(4x+1)y'+(4x+2)y=0; \quad y_1=e^{2x}$$

[exer:5.6.31] $$x^2y''-3xy'+4y=4x^4,\quad y(-1)=7,\quad y'(-1)=-8; \quad y_1=x^2$$

[exer:5.6.32] $$(3x-1)y''-(3x+2)y'-(6x-8)y=0, \quad y(0)=2,\; y'(0)=3; \quad y_1=e^{2x}$$

[exer:5.6.33] $$(x+1)^2y''-2(x+1)y'-(x^2+2x-1)y=(x+1)^3e^x, \quad y(0)=1,\quad y'(0)=~-1$$;

$$y_1=(x+1)e^x$$

[exer:5.6.34] $$x^2y''+2xy'-2y=x^2, \quad y(1)={5\over4},\; y'(1)={3\over2}; \quad y_1=x$$

[exer:5.6.35] $$(x^2-4)y''+4xy'+2y=x+2, \quad y(0)=-{1\over3},\quad y'(0)=-1; \quad y_1={1\over x-2}$$

[exer:5.6.36] Suppose $$p_1$$ and $$p_2$$ are continuous on $$(a,b)$$. Let $$y_1$$ be a solution of

$y''+p_1(x)y'+p_2(x)y=0 \eqno{\rm (A)}$

that has no zeros on $$(a,b)$$, and let $$x_0$$ be in $$(a,b)$$. Use reduction of order to show that $$y_1$$ and

$y_2(x)=y_1(x)\int^x_{x_0}{1\over y^2_1(t)} \exp \left(-\int^t_{x_0}p_1(s)\, ds\right)\,dt$

form a fundamental set of solutions of (A) on $$(a,b)$$.

[exer:5.6.37] The nonlinear first order equation

$y'+y^2+p(x)y+q(x)=0 \eqno{\rm (A)}$

is a Riccati equation. (See Exercise 2.4. [exer:2.4.55}.) Assume that $$p$$ and $$q$$ are continuous.

Show that $$y$$ is a solution of (A) if and only if $$y={z'/z}$$, where

$z''+p(x)z'+q(x)z=0. \eqno{\rm (B)}$

Show that the general solution of (A) is

$y={c_1z'_1+c_2z'_2\over c_1z_1+c_2z_2}, \eqno{\rm (C)}$

where $$\{z_1,z_2\}$$ is a fundamental set of solutions of (B) and $$c_1$$ and $$c_2$$ are arbitrary constants.

Does the formula (C) imply that the first order equation (A) has a two–parameter family of solutions? Explain your answer.

[exer:5.6.38] Use a method suggested by Exercise [exer:5.6.37} to find all solutions. of the equation.

## a

$$y'+y^2+k^2=0$$

## b

$$y'+y^2-3y+2=0$$

## c

$$y'+y^2+5y-6=0$$

## d

$$y'+y^2+8y+7=0$$

## e

$$y'+y^2+14y+50=0$$

## f

$$6y'+6y^2-y-1=0$$

## g

$$36y'+36y^2-12y+1=0$$

[exer:5.6.39] Use a method suggested by Exercise [exer:5.6.37} and reduction of order to find all solutions of the equation, given that $$y_1$$ is a solution.

$$x^2(y'+y^2)-x(x+2)y+x+2=0; \quad y_1=1/x$$

$$y'+y^2+4xy+4x^2+2=0; \quad y_1=-2x$$

$$(2x+1)(y'+y^2)-2y-(2x+3)=0; \quad y_1=-1$$

$$(3x-1)(y'+y^2)-(3x+2)y-6x+8=0; \quad y_1=2$$

$${x^2(y'+y^2)+xy+x^2- {1\over 4}=0; \quad y_1=-\tan x -{1\over 2x}}$$

$${x^2(y'+y^2)-7xy+7=0; \quad y_1=1/x}$$

[exer:5.6.40] The nonlinear first order equation

$y'+r(x)y^2+p(x)y+q(x)=0 \eqno{\rm (A)}$

is the generalized Riccati equation. (See Exercise 2.4. [exer:2.4.55}.) Assume that $$p$$ and $$q$$ are continuous and $$r$$ is differentiable.

Show that $$y$$ is a solution of (A) if and only if $$y={z'/rz}$$, where

$z''+\left[p(x)-{r'(x)\over r(x)}\right] z'+r(x)q(x)z=0. \eqno{\rm (B)}$

Show that the general solution of (A) is

$y={c_1z'_1+c_2z'_2\over r(c_1z_1+c_2z_2)},$

where $$\{z_1,z_2\}$$ is a fundamental set of solutions of (B) and $$c_1$$ and $$c_2$$ are arbitrary constants.