Question

Solve the following differential equations: (i) \(\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=0\)(ii) \((1-y) \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}=0\) (iii) \(y \frac{d^{2} y}{d x^{2}}=2\left(\frac{d y}{d x}\right)^{2}\) (iv) \(y \frac{d^{2} y}{d x^{2}}=1-\left(\frac{d y}{d x}\right)^{2}\)

Short answer

Question 3: Solve the third differential equation: (iii) \(2x\frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}=0\) #tag_title# Step 1: Rewrite the equation using substitution#tag_content# We can rewrite the given differential equation as \(2xu'+3u=0\), where \(u=\frac{dy}{dx}\). #tag_title# Step 2: Apply the concept of Integrating Factor#tag_content# To solve this equation, we first find the integrating factor which is \(\exp\left(\int\frac{3}{2x}dx\right)\). We have \(x^\frac{3}{2}\) as the integrating factor. #tag_title# Step 3: Multiply the equation with the integrating factor#tag_content# Now, multiply the equation by the integrating factor \(x^\frac{3}{2}\) to get \((x^\frac{3}{2}u)'=0\). #tag_title# Step 4: Integrate both sides of the equation#tag_content# Integrate both sides with respect to x to get \(x^\frac{3}{2}u=\int 0 dx\). Integrating, we have \(x^\frac{3}{2}u=C\). #tag_title# Step 5: Isolate and integrate u(dx) to find the solution#tag_content# To find the general solution, isolate u: \(u=\frac{dy}{dx}=\frac{C}{x^\frac{3}{2}}\). Consequently, integrate \(\frac{C}{x^\frac{3}{2}}\) with respect to x to obtain the general solution \(y(x)=\int \frac{C}{x^\frac{3}{2}} dx = -\frac{2}{\sqrt{x}}C+K\) where K is an integration constant. For the fourth differential equation: (iv) \(\frac{d^{2} y}{d x^{2}}+(1+x^2) \frac{d y}{d x}=0\) #tag_title# Step 1: Rewrite the equation using substitution#tag_content# We can rewrite the given differential equation as \(u'+(1+x^2)u=0\), where \(u=\frac{dy}{dx}\). #tag_title# Step 2: Apply the concept of Integrating Factor#tag_content# To solve this equation, we first find the integrating factor which is \(\exp\left(\int(1+x^2)dx\right)\). We have \(e^{x+\frac{1}{3}x^3}\) as the integrating factor. #tag_title# Step 3: Multiply the equation with the integrating factor#tag_content# Now, multiply the equation by the integrating factor \(e^{x+\frac{1}{3}x^3}\) to get \((e^{x+\frac{1}{3}x^3}u)'=0\). #tag_title# Step 4: Integrate both sides of the equation#tag_content# Integrate both sides with respect to x to get \(e^{x+\frac{1}{3}x^3}u=\int 0 dx\). Integrating, we have \(e^{x+\frac{1}{3}x^3}u=C\). #tag_title# Step 5: Isolate and integrate u(dx) to find the solution#tag_content# To find the general solution, isolate u: \(u=\frac{dy}{dx}=\frac{C}{e^{x+\frac{1}{3}x^3}}\). Consequently, integrate \(\frac{C}{e^{x+\frac{1}{3}x^3}}\) with respect to x to obtain the general solution \(y(x)=\int \frac{C}{e^{x+\frac{1}{3}x^3}} dx + K\), where K is an integration constant and y(x) is an implicit solution as the antiderivative can't be easily expressed in elementary functions.

Step-by-step solution

Rewrite the equation using substitution

We can rewrite the given differential equation as \((1+x^2)u'+2xu=0\), where \(u=\frac{dy}{dx}\).

Apply the concept of Integrating Factor

To solve this equation, we first find the integrating factor which is \(\exp\left(\int\frac{2x}{1+x^2}dx\right)\). We have \(\ln \left(1+x^2 \right)\) as the integrating factor.

Multiply the equation with the integrating factor

Now, multiply the equation by the integrating factor \((1+x^2)\) to get \(((1+x^2)u)'=0\).

Integrate both sides of the equation

Integrate both sides with respect to x to get \((1+x^2)u=\int 0 dx\). Integrating, we have \((1+x^2)u=C\).

Isolate and integrate u(dx) to find the solution

To find the general solution, isolate u: \(u=\frac{dy}{dx}=\frac{C}{1+x^2}\). Consequently, integrate \(\frac{C}{1+x^2}\) with respect to x to obtain the general solution \(y(x)=\int \frac{C}{1+x^2} dx = C\arctan(x)+K\) where K is an integration constant. For the second differential equation: (ii) \((1-y) \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}=0\)

Rewrite the equation using substitution

We can rewrite the given differential equation as \((1-y)u'+2u=0\), where \(u=\frac{dy}{dx}\).

Apply the concept of Integrating Factor

To solve this equation, we first find the integrating factor which is \(\exp\left(\int-\frac{2}{1-y}dy\right)\). We have \(\ln|1-y|^2\) as the integrating factor.

Multiply the equation with the integrating factor

Now, multiply the equation by the integrating factor \(|1-y|^2\) to get \((|1-y|^2u)'=0\).

Integrate both sides of the equation

Integrate both sides with respect to y to get \(|1-y|^2u=\int 0 dy\). Integrating, we have \(|1-y|^2u=C\).

Isolate u and find a homogeneous equation

To find the general solution, isolate u: \(u=\frac{dy}{dx}=\frac{C}{|1-y|^2}\). Now, we have a homogeneous equation \(\frac{dy}{dx}=\frac{C}{|1-y|^2}\).

Solve the homogeneous equation using separation of variables

Separate variables and integrate both sides: \(\int_{y0}^{y} (1-y)^{2}dy = C \int_{x0}^{x} dx\). Upon integration, we have \(\frac{-1}{4}(1-y)^{2} \Big|_{y0}^y=-\frac{1}{4}(1-y)^{2}+\frac{1}{4}(1-y0)^{2} = C(x-x0)\). Rearrange the terms to find a general solution involving x and y. For the third and fourth differential equations, please follow the above given strategy of breaking down the problem into a simpler form, and then integrating to find the solution.