Question
Solve the following differential equation
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
✓
Answer
We have,
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
$\Rightarrow\text{dy}=\frac{\cos3\text{x}+\cos2\text{x}}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=\frac{4\cos^2\text{x}-3\cos\text{x}+2\cos^2\text{x}-1}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=(4\cos^2\text{x}-3+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{dy}[2(2\cos^2\text{x}-1)-1+2\cos\text{x}-\sec\text{x}]\text{dx}$
$\Rightarrow\text{dy}(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{ dx}$
Integrating both sides, we get
$\int\text{dy}=\int(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$
hence, $\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$ is the solution to the given differential equation.
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
$\Rightarrow\text{dy}=\frac{\cos3\text{x}+\cos2\text{x}}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=\frac{4\cos^2\text{x}-3\cos\text{x}+2\cos^2\text{x}-1}{\cos\text{x}}\ \text{dx}$
$\Rightarrow\text{dy}=(4\cos^2\text{x}-3+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{dy}[2(2\cos^2\text{x}-1)-1+2\cos\text{x}-\sec\text{x}]\text{dx}$
$\Rightarrow\text{dy}(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{ dx}$
Integrating both sides, we get
$\int\text{dy}=\int(2\cos2\text{x}-1+2\cos\text{x}-\sec\text{x})\text{dx}$
$\Rightarrow\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$
hence, $\text{y}=\sin2\text{x}-\text{x}+2\sin\text{x}-\log|\sec\text{x}+\tan\text{x}|+\text{C}$ is the solution to the given differential equation.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as
10,500 and
9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?
10,500 and9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.
If $\text{x}=10(\text{t}-\sin\text{t}),\text{y}=12(1-\cos\text{t}),$ find $\frac{\text{dy}}{\text{dx}}.$
→Find the angle between two curves $y^2=4 a x$ and $x^2=4 b y$.
→If x and y are connected parametrically by the equation $x=\frac{\sin ^{3} t}{\sqrt{\cos 2 t}}, y=\frac{\cos ^{3} t}{\sqrt{\cos 2 t}}$, without eliminating the parameter, Find $\frac{d y}{d x}$
→For each of the differential equations given in find the general solution:
$(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=1$
→$(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=1$
Using integration, find the area of the region: $\left\{(\text{x},\text{y}) : |\text{x}-1|<\text{y}<\sqrt{5-\text{x}^{2}}\right\}$.
→Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin3\text{x}\text{ on }[0,\pi]$
→$\text{f}(\text{x})=\sin3\text{x}\text{ on }[0,\pi]$