General Solutions of Three First/Second-Order ODEs: 04-BS-1, May 2013, Q2

Question 2: General Solutions of Three First/Second-Order ODEs (a) 7, (b) 7, (c) 6 marks

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Given. Three independent ODEs: (a) first-order linear, (b) first-order separable (Bernoulli with $n=2$), (c) second-order linear homogeneous, constant coefficients.

Find. The general solution of each.

Approach. (a) integrating factor; (b) direct separation of variables; (c) characteristic equation with complex roots.

(a) Integrating factor. Standard form $y'+2xy=2xe^{-x^2}$ has integrating factor $\mu=e^{\int 2x\,dx}=e^{x^2}$. Multiplying through, $$\left(e^{x^2}y\right)'=2xe^{x^2}e^{-x^2}=2x \;\Rightarrow\; e^{x^2}y=x^2+C.$$ $$\Rightarrow\quad y=\boxed{(x^2+C)e^{-x^2}}$$
(b) Separation of variables. Rewrite $\dfrac{dy}{dx}=-2xy^2$. For $y\ne0$, separate: $$\int \frac{dy}{y^2}=-\int 2x\,dx \;\Rightarrow\; -\frac{1}{y}=-x^2+C_1 \;\Rightarrow\; \frac{1}{y}=x^2-C_1.$$ Relabelling the constant, $$\Rightarrow\quad y=\boxed{\dfrac{1}{x^2+C}}\quad\text{(plus the trivial solution } y\equiv 0\text{)}.$$
(c) Characteristic equation. $r^2-2r+3=0\Rightarrow r=\dfrac{2\pm\sqrt{4-12}}{2}=1\pm i\sqrt2$. Complex-conjugate roots give $$\Rightarrow\quad y=\boxed{e^{x}\left(C_1\cos(\sqrt2\,x)+C_2\sin(\sqrt2\,x)\right)}$$

Part	General solution
(a)	$y=(x^2+C)e^{-x^2}$
(b)	$y=\dfrac{1}{x^2+C}$ (plus $y\equiv0$)
(c)	$y=e^{x}\left(C_1\cos(\sqrt2\,x)+C_2\sin(\sqrt2\,x)\right)$

Concept & Theory

Topic. First-order linear (integrating factor), separable, and second-order linear homogeneous ODEs.

Key relations. Integrating factor $\mu(x)=e^{\int P(x)dx}$ for $y'+P(x)y=Q(x)$; separable ODEs $dy/g(y)=f(x)dx$ integrate directly; characteristic roots $\alpha\pm i\beta$ give oscillatory-times-exponential solutions.

Why this works. (a) is exact once multiplied by $\mu=e^{x^2}$ because $\mu$ is chosen precisely so the left side becomes $(\mu y)'$. (b) is a Bernoulli equation with $n=2$; dividing by $y^2$ and separating is equivalent to (but faster than) the standard Bernoulli substitution $v=y^{1-n}=y^{-1}$. (c) follows directly from the linear-ODE superposition principle once the two independent complex-exponential solutions are recombined into real sine/cosine form via Euler's formula.

Common pitfall. In (b), losing the singular solution $y\equiv0$ when dividing by $y^2$; in (a), forgetting to multiply the *entire* equation (including the right side) by the integrating factor before integrating.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 1 (First-Order ODEs, §1.5 Linear ODEs / Bernoulli) and Ch. 2 (§2.2 Homogeneous Linear ODEs).

Question 2 of 8: General Solutions of Three First/Second-Order ODEs

Question 2: General Solutions of Three First/Second-Order ODEs (a) 7, (b) 7, (c) 6 marks