3.6E: Exercises for Section 3.6

Exercise \(\PageIndex{1}\)

\((2x+1)y''-2y'-(2x+3)y=(2x+1)^2; \quad y_1=e^{-x}\)

Exercise \(\PageIndex{2}\)

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

Answer

\(y = ux = \frac{4}{3x^2} - \frac{C_1}{2x} + C_2 x, \quad \text{or} \quad y = \frac{4}{3x^2} + c_1 x + \frac{c_2}{x}.\)

As a byproduct, \(\{x, 1/x\}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{3}\)

\(x^2y''-xy'+y=x; \quad y_1=x\)

Exercise \(\PageIndex{4}\)

\(y''-3y'+2y={1\over1+e^{-x}}; \quad y_1=e^{2x}\)

Answer

\(y = (e^{2x} + e^x) \ln(1 + e^{-x}) + c_1 e^{2x} + c_2 e^x.\)

As a byproduct, \(\{e^{2x}, e^x\}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{5}\)

\(y''-2y'+y=7x^{3/2}e^x; \quad y_1=e^x\)

Exercise \(\PageIndex{6}\)

\(4x^2y''+(4x-8x^2)y'+(4x^2-4x-1)y=4x^{1/2}e^x(1+4x); \quad y_1=x^{1/2}e^x\)

Answer

\(y = e^x \left( 2x^{3/2} + x^{1/2} \ln x + c_1 x^{1/2} + c_2 x^{-1/2} \right).\)

As a byproduct, \(\{ x^{1/2} e^x, x^{-1/2} e^{-x} \}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{7}\)

\(y''-2y'+2y=e^x\sec x; \quad y_1=e^x\cos x\)

Exercise \(\PageIndex{8}\)

\(y''+4xy'+(4x^2+2)y=8e^{-x(x+2)}; \quad y_1=e^{-x^2}\)

Answer

\(y = e^{-x^2} (2e^{-2x} + c_1 + c_2 x).\)

As a byproduct, \(\{e^{-x^2}, x e^{-x^2}\}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{9}\)

\(x^2y''+xy'-4y=-6x-4; \quad y_1=x^2\)

Exercise \(\PageIndex{10}\)

\(x^2y''+2x(x-1)y'+(x^2-2x+2)y=x^3e^{2x}; \quad y_1=xe^{-x}\)

Answer

\(y = \frac{x e^{2x}}{9} + x e^{-x} (c_1 + c_2 x).\)

As a byproduct, \(\{x e^{-x}, x^2 e^{-x}\}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{11}\)

\(x^2y''-x(2x-1)y'+(x^2-x-1)y=x^2e^x; \quad y_1=xe^x\).

Exercise \(\PageIndex{12}\)

\((1-2x)y''+2y'+(2x-3)y=(1-4x+4x^2)e^x; \quad y_1=e^x\)

Exercise \(\PageIndex{13}\)

\(x^2y''-3xy'+4y=4x^4; \quad y_1=x^2\)

Exercise \(\PageIndex{14}\)

\(2xy''+(4x+1)y'+(2x+1)y=3x^{1/2}e^{-x}; \quad y_1=e^{-x}\)

Answer

\(y = e^{-x} \left(x^{3/2} + c_1 + c_2 x^{1/2} \right).\)

As a byproduct, \(\{e^{-x}, x^{1/2} e^{-x}\}\) is a fundamental set of solutions of the complementary equation.

Exercise \(\PageIndex{15}\)

\(xy''-(2x+1)y'+(x+1)y=-e^x; \quad y_1=e^x\)

Exercise \(\PageIndex{16}\)

\(4x^2y''-4x(x+1)y'+(2x+3)y=4x^{5/2}e^{2x}; \quad y_1=x^{1/2}\)

Exercise \(\PageIndex{17}\)

\(x^2y''-5xy'+8y=4x^2; \quad y_1=x^2\)

Exercise \(\PageIndex{18}\)

\(xy''+(2-2x)y'+(x-2)y=0; \quad y_1=e^x\)

Answer

\(y = e^x \left( -\frac{C_1}{x} + C_2 \right) \text{ is the general solution, and } \{e^x, e^x/x\} \text{ is a fundamental set of solutions.}\)

Exercise \(\PageIndex{19}\)

\(x^2y''-4xy'+6y=0; \quad y_1=x^2\)

Exercise \(\PageIndex{20}\)

\(x^2(\ln |x|)^2y''-(2x \ln |x|)y'+(2+\ln |x|)y=0; \quad y_1=\ln |x|\)

Exercise \(\PageIndex{21}\)

\(4xy''+2y'+y=0; \quad y_1=\sin \sqrt{x}\)

Exercise \(\PageIndex{22}\)

\(xy''-(2x+2)y'+(x+2)y=0; \quad y_1=e^x\)

Answer

\(y = \left( \frac{C_1 x^3}{3} + C_2 \right) e^x \text{ is the general solution, and } \{e^x, x^3 e^x\} \text{ is a fundamental set of solutions.}\)

Exercise \(\PageIndex{23}\)

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

Exercise \(\PageIndex{24}\)

\(x^2y''-2xy'+(x^2+2)y=0; \quad y_1=x \sin x\)

Exercise \(\PageIndex{25}\)

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

Exercise \(\PageIndex{26}\)

\(4x^2(\sin x)y''-4x(x\cos x+\sin x)y'+(2x\cos x+3\sin x)y=0; \quad y_1=x^{1/2}\)

Exercise \(\PageIndex{27}\)

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

Exercise \(\PageIndex{28}\)

\((2x+1)xy''-2(2x^2-1)y'-4(x+1)y=0; \quad y_1=1/x\)

Exercise \(\PageIndex{29}\)

\((x^2-2x)y''+(2-x^2)y'+(2x-2)y=0; \quad y_1=e^x\)

Exercise \(\PageIndex{30}\)

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

Answer

\(y = e^{2x} \left( \frac{C_1 x^2}{2} + C_2 \right) \text{ is the general solution, and } \{e^{2x}, x^2 e^{2x}\} \text{ is a fundamental set of solutions.}\)

Exercise \(\PageIndex{31}\)

\(x^2y''-3xy'+4y=4x^4,\quad y(-1)=7,\quad y'(-1)=-8; \quad y_1=x^2\)

Exercise \(\PageIndex{32}\)

\((3x-1)y''-(3x+2)y'-(6x-8)y=0, \quad y(0)=2,\; y'(0)=3; \quad y_1=e^{2x}\)

Answer

\(y = 2e^{2x} - x e^{-x}.\)

Exercise \(\PageIndex{33}\)

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

Exercise \(\PageIndex{34}\)

\(x^2y''+2xy'-2y=x^2, \quad y(1)={5\over4},\; y'(1)={3\over2}; \quad y_1=x\)

Answer

\(y = \frac{x^2}{4} + x.\)

Exercise \(\PageIndex{35}\)

\((x^2-4)y''+4xy'+2y=x+2, \quad y(0)=-{1\over3},\quad y'(0)=-1; \quad y_1={1\over x-2}\)

Exercise \(\PageIndex{36}\)

Suppose \(p_1\) and \(p_2\) are continuous on \((a,b)\). Let \(y_1\) be a solution of

\[y''+p_1(x)y'+p_2(x)y=0 \tag{A} \]

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

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

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

Exercise \(\PageIndex{37}\)

The nonlinear first order equation

\[y'+y^2+p(x)y+q(x)=0 \tag{A} \]

is a Riccati equation. Assume that \(p\) and \(q\) are continuous.

1. Show that \(y\) is a solution of (A) if and only if \(y={z'/z}\), where \[z''+p(x)z'+q(x)z=0. \tag{B} \]
2. Show that the general solution of (A) is \[y={c_1z'_1+c_2z'_2\over c_1z_1+c_2z_2}, \tag{C} \] where \(\{z_1,z_2\}\) is a fundamental set of solutions of (B) and \(c_1\) and \(c_2\) are arbitrary constants.
3. Does the formula (C) imply that the first order equation (A) has a two–parameter family of solutions? Explain your answer.

Exercise \(\PageIndex{38}\)

Use a method suggested by Exercise 3.6E.37 to find all solutions of the equation.

1. \(y'+y^2+k^2=0\)
2. \(y'+y^2-3y+2=0\)
3. \(y'+y^2+5y-6=0\)
4. \(y'+y^2+8y+7=0\)
5. \(y'+y^2+14y+50=0\)
6. \(6y'+6y^2-y-1=0\)
7. \(36y'+36y^2-12y+1=0\)

Exercise \(\PageIndex{39}\)

Use the method of reduction of order to find all solutions of the equation, given that \(y_1\) is a solution.

1. \(x^2(y'+y^2)-x(x+2)y+x+2=0; \quad y_1=1/x\)
2. \(y'+y^2+4xy+4x^2+2=0; \quad y_1=-2x\)
3. \((2x+1)(y'+y^2)-2y-(2x+3)=0; \quad y_1=-1\)
4. \((3x-1)(y'+y^2)-(3x+2)y-6x+8=0; \quad y_1=2\)
5. \({x^2(y'+y^2)+xy+x^2- {1\over 4}=0; \quad y_1=-\tan x -{1\over 2x}}\)
6. \({x^2(y'+y^2)-7xy+7=0; \quad y_1=1/x}\)

Exercise \(\PageIndex{40}\)

The nonlinear first order equation

\[y'+r(x)y^2+p(x)y+q(x)=0 \tag{A} \]

is the generalized Riccati equation. Assume that \(p\) and \(q\) are continuous and \(r\) is differentiable.

1. Show that \(y\) is a solution of (A) if and only if \(y={z'/rz}\), where \[z''+\left[p(x)-{r'(x)\over r(x)}\right] z'+r(x)q(x)z=0. \tag{B} \]
2. Show that the general solution of (A) is \[y={c_1z'_1+c_2z'_2\over r(c_1z_1+c_2z_2)}, \nonumber \] where \(\{z_1,z_2\}\) is a fundamental set of solutions of (B) and \(c_1\) and \(c_2\) are arbitrary constants.